ABSTRACT
Title of dissertation: SPIN INJECTION AND DETECTION
IN COPPER SPIN VALVE STRUCTURES
Samir Garz?on, Doctor of Philosophy, 2005
Dissertation directed by: Richard A. Webb
Department of Physics
We report measurements of spin injection and detection in a mesoscopic copper
wire from which the electron spin relaxation time and the spin current polarization
in copper can be found. Spin injection is realized by applying a voltage to drive a
current from a ferromagnet into the normal metal, while spin detection is done using
transport measurements. Precession of the spin of the injected electrons due to an
external magnetic field is also studied. The existence of a previously unobserved spin
signal which vanishes at low temperatures but increases nonlinearly above 100oK
is reported and a possible explanation for its origin, based on interfacial spin-flip
scattering, is suggested. Multiple cross checks to test the possibility of artifacts as
an origin of this signal are discussed.
An alternative spin detection method using magnetic force microscopy (MFM)
is also studied. This method measures the magnetic field produced by the injected
spins directly, so the spin coherence length and the spin current polarization can be
extracted directly without the need of a particular transport model, avoiding issues
like contact resistance and interface scattering. The MFM method can also be useful
for measuring the spin polarization of currents in semiconductors and semiconductor
heterostructures, which is important for the development of spintronics.
SPIN INJECTION AND DETECTION
IN COPPER SPIN VALVE STRUCTURES
by
Samir Garz?on
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2005
Advisory Commmittee:
Professor Richard A. Webb, Chair/Advisor
Professor J. Robert Anderson
Assistant Professor Michael S. Fuhrer
Professor Romel D. Gomez
Professor Ellen D. Williams
c? Copyright by
Samir Garzon
2005
ACKNOWLEDGEMENTS
First of all I want to give thanks to my family. To my mom and dad, who,
from a distance, provided constant support and encouragement. They helped me
get through the toughest moments. To my sister, Cosh, who was always able to put
a smile on my face. And to my brother, who was always there when I needed him.
I also want to give thanks to Professor Webb for his help in the lab and
his patience at times when research progress was slow, but over all, for allowing
me the freedom to work on my own. I want to thank my labmates, Chen and
Tarek, for their company and conversations, which made the lab feel like a second
home. I want to give special thanks to Chen for his encouragement during the last
year. The dissertation committee members, Professors Anderson, Fuhrer, Gomez,
and Williams, also deserve many thanks for having taken the time to understand
my work and for giving me valuable suggestions. Professors Fuhrer, Gomez, and
Williams deserve additional thanks for allowing me to use the MFM?s in their labs.
I want to give special thanks to Igor for being of great help during the last
year. He gave very valuable suggestions, facilitated useful discussions with Dr. Mark
Johnson from NRL, and encouraged me to go on during some tough moments.
ii
I also want to thank the staff in the Center for Superconductivity Research,
who was always helpful and kept things running smoothly, and the technicians in
charge of maintenance, especially those in charge of the HVAC, since they were
always willing to help, and kept the system running (or at least, walking). I also
want to thank the Physics Department staff for their help. In particular I want
to thank Lorraine, who helped keep the lab moving whenever we had technical
problems, and quickly relocated us once Webb?s lab moved out, and to Jane, who
was always willing to help, and made dealing with paperwork a much easier task.
Finally, I want to thank all my friends, inside and outside the Physics Depart-
ment, for making the time outside the lab enjoyable. Without them, these five and
a half years in College Park wouldn?t have been worthwhile. I want to give special
thanks to my roommates, Juan and Luis, who were my second family, and to Paiki,
for her time, company, and patience.
iii
TABLE OF CONTENTS
List of Figures vii
List of Tables xvii
1 Introduction 1
1.1 Spintronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Spin injection and detection: transport . . . . . . . . . . . . . . . . . 8
1.3 Magnetic force microscopy . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Theory of spin transport and decoherence 15
2.1 Single spin coherence time, spin dephasing, and spin relaxation times 15
2.2 Spin relaxation mechanisms . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Elliot-Yafet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Magnetic impurity scattering . . . . . . . . . . . . . . . . . . 20
3 Spin Transport 22
3.1 Spin transport: diffusion equation . . . . . . . . . . . . . . . . . . . . 24
3.1.1 Spin transport in one dimension . . . . . . . . . . . . . . . . . 28
3.1.2 Interfacial spin transport . . . . . . . . . . . . . . . . . . . . . 28
3.2 Spin valve structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.1 Cross geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.2 Nonlocal geometry . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.3 Contact resistance . . . . . . . . . . . . . . . . . . . . . . . . 49
iv
3.3 Effect of leakage currents . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Hanle effect: spin precession . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Interface spin-flip scattering . . . . . . . . . . . . . . . . . . . . . . . 54
3.5.1 Nonlocal geometry spin valve with interfacial spin-flip scattering 54
4 Sample fabrication and experimental setup 61
4.1 Sample fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Spin Valve Measurements 74
5.1 Cross geometry configuration . . . . . . . . . . . . . . . . . . . . . . 74
5.2 Nonlocal geometry configuration . . . . . . . . . . . . . . . . . . . . . 78
5.2.1 Length dependence of the nonlocal resistance . . . . . . . . . 79
5.2.2 Spin precession: Hanle effect . . . . . . . . . . . . . . . . . . . 83
5.2.3 Symmetric component of the nonlocal resistance . . . . . . . . 89
5.2.4 Length dependence of RS . . . . . . . . . . . . . . . . . . . . 100
5.2.5 Temperature dependence of RS for a control sample . . . . . . 103
5.2.6 Heating and thermoelectric effect . . . . . . . . . . . . . . . . 104
5.2.7 Geometric effects . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2.8 Current leakage . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2.9 Lock-in tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6 Spin injection and magnetic force microscopy detection I: Introduction 116
6.1 Scanning probe microscope operation and model . . . . . . . . . . . . 117
v
6.2 Spin injection and MFM detection . . . . . . . . . . . . . . . . . . . 121
6.3 Increasing the resolution of the cantilever . . . . . . . . . . . . . . . . 129
6.4 Resonant MFM operation . . . . . . . . . . . . . . . . . . . . . . . . 133
6.5 Force sensitivity: thermal noise . . . . . . . . . . . . . . . . . . . . . 136
6.6 Cantilever Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7 Spin injection and magnetic force microscopy detection II: Experiment 146
7.1 Principle of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.3 Flat sample fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.4 Electric potential balancing . . . . . . . . . . . . . . . . . . . . . . . 155
7.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8 Summary and Conclusions 168
8.1 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.2 MFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
A Magnetic field properties, Green?s functions, and convolutions 173
A.1 Magnetic field generated by an in-plane current . . . . . . . . . . . . 173
A.2 Magnetic field generated by an in-plane magnetization . . . . . . . . 178
A.3 2-D Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
A.4 1-D Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Bibliography 182
vi
LIST OF FIGURES
1.1 (a) Schematic of a GMR structure for a magnetic reading head. The
two ferromagnetic layers (F1 and F2) separated by a nonmagnetic
metal (N) are shown, together with the leads for applying a sensing
current. (b) Schematic of a Magnetic Tunnel Junction (MTJ) used
for Magnetic Random Access Memories (MRAM?s). . . . . . . . . . . 5
1.2 Spin dependent density-of-states diagrams of a ferromagnet placed
in contact with a nonmagnetic metal when (a) no current is being
injected, and (b) a current is driven between the two metals. The
dashed line represents the average electrochemical potential ?0. . . . 9
1.3 Position dependence of the electrochemical potential ? for both spin
channels near a F-N interface in the presence of electric current. . . . 10
1.4 (a) Sample geometry. Shaded regions represent two Co ferromagnets
separated by a distance L, while the solid black region represents
the Cu line. Ti and Bi are nonmagnetic measurement leads. (b) ??
for spin up and spin down electrons in injector (left), normal metal
(center), and detector (right) as a function of position. Detector ??
for parallel (solid) and antiparallel (dashed) magnetizations is shown. 12
3.1 Schematic of a spin valve device showing magnetic contacts (gray),
normal metal wires (black), and normal metal leads (white) for cur-
rent transport and voltage measurement. . . . . . . . . . . . . . . . . 23
vii
3.2 Schematic of the transport across a nonmagnetic-ferromagnet inter-
face in the case of conserved spin. The two transport channels (spin
? and spin ?) and their associated interface resistances are shown. . . 30
3.3 Schematic of a spin valve device with magnetic contacts (gray), nor-
mal metal wires (black), and normal metal leads (white) for current
transport and voltage measurement, in the cross geometry configura-
tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Resistance given by equation (3.20) as a function of separation L
between magnetic contacts for ?1?2 =?? (aligned magnetizations of
F1 and F2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Electrochemical potential for both spin channels and both aligned
(solid line) and anti-aligned (dashed line) magnetizations of F1 and F2. 36
3.6 Spin component of the electrochemical potential for aligned (solid
line) and anti-aligned (dashed line) magnetizations of F1 and F2. . . . 37
3.7 Electrochemical potential for both spin channels and both aligned
(solid line) and anti-aligned (dashed line) magnetizations of F1 and
F2 for spin current polarizations P1 = P2 = P of (a) 5%, (b) 10%,
and (c) 20% for a contact resistance of 5 ?. . . . . . . . . . . . . . . 41
3.8 Spin electrochemical potential for both aligned (solid line) and anti-
aligned (dashed line) magnetizations of F1 and F2 for spin current
polarizations P1 = P2 = P of (a) 5%, (b) 10%, and (c) 20% for a
contact resistance of 5 ?. . . . . . . . . . . . . . . . . . . . . . . . . . 43
viii
3.9 Schematic of a spin valve device for nonlocal measurements. Gray
regions represent the magnetic contacts, black ones the normal metal
wire, and white regions the connecting leads. . . . . . . . . . . . . . . 44
3.10 Nonlocal resistance for a current of 100 ?A in the presence of leakage
currents through regions 4, 5, and 7 in Fig. 3.9. The horizontal axis
gives the leakage current through each of the leads. . . . . . . . . . . 51
3.11 Diagrams representing the scattering at the interface together with
phenomenological coefficientsRthatrepresent the interfaceresistance
of each spin channel. (a) Majority carriers of each region are aligned
and have spin ?. (b) Majority carriers of the ferromagnet have spin
? while majority carriers of the nonmagnetic metal have spin ?. (c)
Majority carriers of each region are aligned and have spin ?. (d) Ma-
jority carriers of the ferromagnet have spin ? while majority carriers
of the nonmagnetic metal have spin ?. . . . . . . . . . . . . . . . . . 55
4.1 Schematic of the e-beam lithography and thermal evaporation pro-
cess for sample fabrication. A side view of the wafer is shown. (a)
Substrate and PMMA bilayer. (b) Exposure of the PMMA to the
e-beam. (c) Developing (or removal) of the exposed PMMA. (d) Ion
milling of the substrate for cleaning and increasing adhesion. (e)
Thermal evaporation of metal. (f) Lift-off of the remaining PMMA
and metallization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
ix
4.2 SEM micrographs of the typical large field samples design. (a) Fo-
cusing and alignment spot, together with additional wiring at a field
scale of 120 ?m. (b) Bonding pads and additional wiring at a field
scale of 2mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 SEM micrographs of finished spin-valve samples. (a) Cross geome-
try Py-Cu sample.(b) Typical nonlocal resistance Co-Cu sample. (c)
Typical contact between magnetic and non-magnetic levels for a Py-
Cu sample. (d) Sample with many magnetic contacts for measuring
the length dependence of the nonlocal resistance. . . . . . . . . . . . 68
5.1 R vs Bbardbl for sample sam2b8s1 at 4.2 K. This sample has transparent
Py-Cu contacts. The inset shows a schematic of the sample layout
and measurement configuration. . . . . . . . . . . . . . . . . . . . . . 75
5.2 R vs Bbardbl for sample sam2b8s1 at 77 K. This sample has transparent
Py-Cu contacts. The inset shows a schematic of the sample layout
and measurement configuration. . . . . . . . . . . . . . . . . . . . . . 78
5.3 Nonlocal resistance switching (RA =
bracketleftBig
R??NL ?R??NL
bracketrightBig
/2) as a func-
tion of the separation L between injector and detector for sample
sam4b1s4, with transparent Py-Cu contacts at 4.2 K. . . . . . . . . . 80
5.4 Nonlocal resistance switching (RA) as a function of injector-detector
separationL for(a)theaverage ofsamples sam10b3p1andsam10b3p3,
and (b) sample sam12b1p2, with transparent Py-Cu contacts at 4.2 K. 82
x
5.5 (a) Hanle effect measurement of sample s sam11b2y1p2 at 4.2 K for
aligned injector and detector separated by 425 nm; the nonlocal re-
sistance is shown as a function of the out-of-plane magnetic field B?.
A fit to the data is also shown, together with the fitting parameters
?=22.2 ps and?P1P2=5.5%. (b) Schematic of the Hanle effect exper-
iment. The nonlocal resistance, defined as RNL = VT2?N2/IT1?N1 for
either aligned or anti-aligned magnetizations of injector and detector
is measured as a function of the out-of-plane magnetic field B?. . . . 84
5.6 Hanle effect measurements and fits for two different injector-detector
separations (sample s sam11b2y3p2) at 4.2 K. (a) 867 nm, aligned
injector and detector (??). (b) 1110 nm, ??. (c) 867 nm, ??. (d)
1110 nm, ??. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.7 Hanleeffect measurement forsample s sam12p2 with injector-detector
separation of 717 nm at 4.2 K, for (a) 19.6 ?A bias and (b) 9.8 ?A
bias. Notethe largevalue of the spin current polarization, about three
times larger than for other samples, but a comparable spin diffusion
length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.8 Temperature dependence of the nonlocal resistance for aligned (??)
and anti-aligned (??) magnetizations of injector and detector for sam-
ple s sam11b2y1p2 with an injector-detector separation of 425 nm.
Dashed lines show the 4.2 K values. . . . . . . . . . . . . . . . . . . . 90
xi
5.9 Temperature dependence of(a) theresistivity of Co, (b)the resistivity
of Cu, and (c) the contact resistance. . . . . . . . . . . . . . . . . . . 91
5.10 Symmetric and anti-symmetric components of the nonlocal resistance
as defined in equation (5.2), for the data shown in Fig. 5.8. . . . . . . 92
5.11 Temperature dependence of PA and PS together with fits to P0(1?
?T3/2) (with P0=5.5% and ?=8.4?10?5K?3/2) and a Fermi function
B/(1+ expT0/T) with an activation temperature of T0=1227K. . . . 97
5.12 Injector-detector separation (L) dependence of the symmetric part
of the nonlocal resistance RS for transparent samples with (a) Py-
Cu interfaces (s sam10b3) and (b) Co-Cu interfaces (s sam10b1p1)
at room temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.13 Injector-detector separation (L) dependence of the symmetric part of
the nonlocal resistance RS for transparent samples with either Py-Cu
or Cu-Cu interfaces. Measurements for samples with Py injector and
detector (Py,Py), Py injector and Cu detector (Py,Cu), Cu injector
and Py detector (Cu,Py), and Cu injector and detector (Cu,Cu) are
shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.14 Comparison of the temperature dependence of RS for a sample with
magnetic injector and detector, and a control sample with nonmag-
netic Ti electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
xii
5.15 (a)Schematic ofSeebeck effect with two dissimilar metals. (b) Schematic
of a typical sample to illustrate the possibility of measuring heating-
related voltages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.16 Differential resistance R as a function of the DC bias current for AC
bias currents of 672 nA (triangles) and 1344 nA (squares), together
with linear fits. The solid (dashed) line is the result of the fit to the
672 nA (1344 nA) data. . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.17 Temperature dependence of RS for four different bias conditions as
indicated in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.18 Two dimensional schematic of the current lines andequipotential lines
in a cross geometry when the current flows between adjacent arms. . 111
5.19 (a) Schematic of the sample with numbered leads. (b) Circuit model
schematic of sample plus wiring. The numbers in the labels of the
components are those illustrated in (a). Resistances are in units of
Ohms and capacitances in units of Farads, with the additional mod-
ifiers pico (p), kilo (k), and giga (g). . . . . . . . . . . . . . . . . . . . 114
6.1 Schematic of a scanning probe microscope (MFM, EFM, or AFM)
with a fixed vibrating cantilever and movable sample stage. . . . . . . 118
6.2 Schematic for spin injection and MFM detection. A current driven
along the copper wire (white) is forced to enter a ferromagnetic sec-
tion (black)of the wire, as a cantilever with a magnetized tip scans
close to the ferromagnetic-nonmagnetic metal interface. . . . . . . . . 122
xiii
6.3 Calculated excess magnetization?M (in the x direction) as a function
of y for a 1.2 ?m wide, 50 nm thick wire carrying a current of 9.1
mA. Realistic values of the resistivity and spin relaxation lengths of
both metals have been assumed for the calculation. . . . . . . . . . . 125
6.4 (a) Magnetic field in the z direction as a function of x and y at a
height of 200 nm above a 1.2 ?m wide, 51 nm thick wire carrying a
current of 9 mA, due to the excess magnetization shown in Fig. 6.3.
(b) First and (c) second derivatives of Bz with respect to z. . . . . . . 128
6.5 High resolution tips with small cobalt covering near the apex. . . . . 131
6.6 Comparison of the response of the normal MFM and high resolution
probes to the field produced by a straight current carrying line. The
high resolution (normal) probe response is shown with a solid black
(gray) line, while the ideal response to a point probe is shown with a
dashed black line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.7 Comparison of the response of the normal MFM phase method and
the resonant method to the field produced by a straight current car-
rying line. The resonant method (phase method) response is shown
with a solid (dashed) line. . . . . . . . . . . . . . . . . . . . . . . . . 136
xiv
6.8 (a) Position dependence of the magnetic signal measured by the can-
tilever (diamonds) at a scan height z=300 nm above the sample. Since
the field is independent of y, only the x dependence is shown. The
solid line shows the convolution of the expected signal [shown in (b)]
with the 1D convolution factor of the tip [shown in the inset of (a)]. . 141
6.9 Measured cantilever noise as a function of bandwidth, with linear fit. 144
7.1 Diagram of the experimental setup. Three lock-in amplifiers are used
to detect the cantilever response at different frequencies, to indepen-
dently measure the fields produced by the magnetic material, the
current, and the excess magnetization from spin injection. . . . . . . 150
7.2 Flat sample fabrication steps. (a) A PMMA mask is defined on top
of a uniform Py film. (b) Ion milling is used to remove unprotected
Py. (c) The hole left by ion milling is filled up with Cu. (d) Second
PMMA mask is defined. (e) Ion milling is used to trim the Cu-Py-Cu
line to the desired width. (f) Final result. . . . . . . . . . . . . . . . . 154
7.3 Schematic of the sample layout for balancing the voltage on the can-
tilever, which is connected to the point with voltage V0 on the left
wire. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.4 Topography of the wire near the Cu-Py-Cu interfaces. . . . . . . . . . 157
7.5 Measured cantilever response to the magnetic field produced by the
current at a frequency of 60.8 kHz. . . . . . . . . . . . . . . . . . . . 158
xv
7.6 Measured cantilever response to the field produced by the magnetic
injector at a frequency of 4.32kHz. . . . . . . . . . . . . . . . . . . . 159
7.7 Measured cantilever response at a frequency of 56.8 kHz and 15 de-
grees phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.8 Measured cantilever response at a frequency of 56.8 kHz and 105
degrees phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.9 V1?0 generated by equation (7.8). . . . . . . . . . . . . . . . . . . . . . 165
xvi
LIST OF TABLES
4.1 Device characteristics. The number of magnetic contacts is given un-
der ?Configuration?, where ?2,2? means two contacts separated by
a large distance from the next two contacts. The magnetic mate-
rial used for each device is specified with a check under either ?Py?
(Permalloy) or ?Co? (Cobalt). If an oxide barrier is used between
ferromagnetic and nonmagnetic metals, the pressure of O2 during
oxidation is given under ?Al2O3?. . . . . . . . . . . . . . . . . . . . . 69
4.2 Device lengths. The abbreviated column titles are ?Thickness? of the
ferromagnetic material, ?Width? of the ferromagnetic material, ?Cu
Thickness?, ?Cu Width?, and ?L?, the separation between contacts.
In the case of many magnetic contacts the separations of consecutive
contacts are given in the footnotes. . . . . . . . . . . . . . . . . . . . 70
5.1 Bias parameters. The first column contains the applied voltage bias,
the second column the AC current bias, and the last one the DC
current bias. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
xvii
Chapter 1
Introduction
This thesis reports measurements of spin injection and detection in a mesoscopic
copper wire from which the electron spin relaxation time and the spin current po-
larization in copper can be found. Spin injection is realized by applying a voltage
to drive a current from a ferromagnet into the normal metal, while spin detection is
done using transport measurements. Precession of the spin of the injected electrons
due to an external magnetic field is also studied. The existence of a previously un-
observed spin signal which vanishes at low temperatures but increases nonlinearly
above 100K is reported and a possible explanation for its origin, based on interfa-
cial spin-flip scattering, is suggested. Multiple cross checks to test the possibility of
artifacts as an origin of this signal are discussed.
An alternative spin detection method using magnetic force microscopy (MFM)
is also proposed. This method measures the magnetic field produced by the injected
spins directly, so the spin coherence length and the spin current polarization can be
extracted directly without the need of a particular transport model, avoiding issues
like contact resistance and interface scattering. The MFM method, as opposed to
the transport method [1], can also be useful for measuring the spin polarization of
1
currents in semiconductors and semiconductor heterostructures, which is important
for the development of spintronics.
The outline of this thesis is as follows. Chapter 1 defines the concept of spin-
tronics and motivates the idea of studying spin injection and detection by providing
a brief overview of the field. The essence of spin injection and detection via transport
is discussed, as well as the possibility of using MFM detection. Chapter 2 introduces
concepts such as single electron coherence time, spin relaxation time, and spin de-
phasing time, and gives a brief overview of the spin scattering mechanisms relevant
for conduction electrons in metals. Chapter 3 discusses spin-dependent transport in
metallic systems, considering the cases of transparent and resistive interfaces, and
local and nonlocal measurement configurations. Additionally the spin precession
of electrons is studied and the effect of interface spin flip scattering is considered.
Chapter 4 explains the details of sample fabrication and experimental setup. The re-
sults and analysis of the spin valve measurements are presented in chapter 5 together
with relevant cross-checks to clarify the role of geometrical effects, and heating and
thermoelectric effect. This finishes the first part of the thesis, which explores spin
injection and detection via transport in spin valve structures. The following two
chapters discuss spin detection using MFM. Chapter 6 introduces scanning probe
microscopy and its limitations, discusses calibration procedures for the cantilevers,
and shows some of the improvements that were made in the resolution and sensi-
tivity of the MFM. Then, chapter 7 shows the additional improvements in signal
detection and sample design that were made, and the null results of the measure-
2
ments. Finally, conclusions and possible directions for future work are presented in
chapter 8.
1.1 Spintronics
Simply stated, electronics is to electric charge as spintronics is to electron spin.
Spintronics, therefore, studies the spins of electrons and their interaction with the
surrounding solid state system. The essential quantities are the population of elec-
trons in different spin states, and their relative phase, which gives the degree of
coherence of the spin populations. Creating a spin-polarized population of electrons
in a nonmagnetic material involves perturbing the system from equilibrium. This
can be done using optical techniques in which, by shining circularly polarized light
on a particular region, a nonequilibrium spin population is created by the transfer
of angular momentum [2]. Equivalently, by injecting a current from a ferromagnetic
material, which has a net spin polarization of the conduction electrons, a nonequi-
librium spin population can also be created [3]. The spin polarization diffuses away
from the injection region, so the spin accumulation is given by a balance of the spin
being produced within a region, the spin leaving (or entering) the region, and the
rate at which the nonequilibrium spin population relaxes and dephases. Typical
spin-independent momentum relaxation mechanisms like phonons and boundary or
impurity scattering can cause spin relaxation due to spin-orbit coupling which mixes
the two spin channels [4, 5]. Additionally, spin-dependent scattering off magnetic
impurities also contributes to momentum and spin relaxation. Hence, in order to
exploit electronic spin for the fabrication of novel devices, it is essential to measure
3
the spin relaxation times and the magnitude of the spin polarization, and to find
ways to enhance them.
The single most important application of spintronics is the use of Giant Mag-
neto Resistive (GMR) effects for magnetic reading heads. All of the currently avail-
able hard drives with GByte storage capacities use GMR magnetic heads for the
readout of stored information. The principle of operation is based on the concept
of a spin valve. Simply stated, a spin valve is a structured composed of layers of
magnetic and nonmagnetic materials, such that the device resistance depends on
the relative alignment of the magnetization of the magnetic layers. Figure 1.1(a)
shows a simple schematic of a GMR structure used for magnetic reading heads.
Two ferromagnetic films (F1 and F2) are separated by a thin layer of a nonmagnetic
metal (N). The magnetization of the top magnetic electrode is pinned as shown
in Fig. 1.1(a) by means of an additional antiferromagnetic layer (not shown in the
schematic). The magnetization of the lower magnetic electrode is free to rotate in
the plane of the film. As the magnetic reading head is scanned over the surface
of a magnetic storage disk, the magnetization of F1 rotates in response to the field
produced by each of the bits in the magnetic disk. By applying a sensing current
in the plane of the films (CIP) and measuring the voltage between the two leads,
the state of the bit can be detected since the resistance between the leads depends
on the alignment of the magnetization of the two ferromagnetic films. Another
important spintronic application is a Magnetic Tunnel Junction (MTJ), which can
be used as a memory device to store a bit of information. A schematic of such a
4
Current perpendicular to plane (CPP) geometry
LeadLead
(b) MTJ for MRAM
Sense CurrentSense Current
Lead
Lead
Current in plane (CIP) geometry
(a) GMR Magnetic Reading Head
SenseCurrent
SenseCurrent2I
F1
F2
N
1F
F
Figure 1.1: (a) Schematic of a GMR structure for a magnetic reading head. The two
ferromagnetic layers (F1 and F2) separated by a nonmagnetic metal (N) are shown,
together with the leads for applying a sensing current. (b) Schematic of a Magnetic
Tunnel Junction (MTJ) used for Magnetic Random Access Memories (MRAM?s).
device is shown in Fig. 1.1(b). The structure is similar to that of GMR, except that
the nonmagnetic material is replaced by an insulator, and the sensing current is
driven perpendicular to the plane of the films (CPP). Depending on the state of the
magnetization of F1, either parallel or antiparallel to F2, the measured resistance
between the leads is different. Since no power is necessary to keep the memory
state of the system, such a device is referred to as being nonvolatile, meaning that
no information is lost after power-off. By fabricating arrays of such structures, it
is possible to fabricate a Magnetic Random Access Memory (MRAM) which could
replace actual RAM?s used in everyday computers since, being nonvolatile, they
5
require no turn-on time. The speed at which this devices can operate, their power
consumption, and the maximum storage density they can achieve depend on the
magnitude of the resistance change between the parallel and antiparallel configura-
tions. The size of this resistance change in GMR structures and MTJ?s is directly
related to the degree of spin current polarization that is either injected into the
normal metal (GMR) or that tunnels across the insulating barrier (MTJ?s), which
is very sensitive to interface effects. Therefore studying the spin transport across
ferromagnetic-nonmagnetic (F-N) and insulating interfaces can be important for the
development of hard drives and MRAM?s that are faster, more power efficient, and
have larger storage density than the ones currently available.
The basis for understanding spin-polarized transport is due to Mott [6, 7], who
realized that the electrical current in ferromagnets could be expressed as the sum of
two independent and unequal parts for two different spin projections implying that
the current is spin-polarized. This idea of a two-current model was extended by
Campbell, Fert, and Valet [8, 9, 10], and provides the explanation for different kinds
of magnetoresistive effects1 such as giant magnetoresistance (GMR) and tunnelling
magnetoresistance (TMR), key elements in applications such as magnetic harddrives
and nonvolatile magnetic random access memory (MRAM) [11, 12].
The interest to study spin accumulation, one of the key elements in spintronic
applications [13], is not limited to novel spin-based devices. Spin injection can also
be used as a sensitive spectroscopic tool to study fundamental properties such as the
1Changes in the resistance in ferromagnetic multi-layer structures depending on the alignment
of the magnetizations of the different layers.
6
pairing symmetry of unconventional superconductors [14, 15], Skyrmion excitations
in the quantum Hall regime [16, 17], and spin-charge separation in non-Fermi liquids
[18, 19].
Spin accumulation in metals was first demonstrated by Johnson and Silsbee
[3], who injected a spin polarized current from permalloy into bulk aluminum and
detected the nonequilibrium spin polarization with a second permalloy contact at a
distance L from the injector. By fabricating samples with different L and by per-
forming spin precession measurements, they were able to extract information on the
degree of spin polarization and the spin relaxation length for aluminum. Johnson
extended these studies to thin gold films [20, 21]. Later, Jedema et. al. [22] applied
Johnson?s injection technique to mesoscopic devices and measured the spin relax-
ation length and the spin polarization in copper. However, their geometry was not
optimal, the data was not very reproducible, and the analysis of the results ques-
tionable. In a later improved experiment Jedema et. al. [23] studied injection into
aluminum through a tunnel barrier and used spin precession to extract the degree of
spin polarization and the spin relaxation length in a mesoscopic aluminum wire. In
this thesis, spin injection and detection measurements in copper mesoscopic wires
are performed using both the cross geometry configuration reported by Jedema et.
al. [22] and a completely nonlocal configuration. Experiments with both transpar-
ent and resistive interfaces are discussed, and spin precession with tunnel barriers is
studied. Additionally, the temperature dependence of the spin signal is measured,
and the appearance of a previously unobserved feature, consistent with interfacial
7
spin-flip scattering, is reported.
1.2 Spin injection and detection: transport
A nonequilibrium spin polarization in a nonmagnetic material can be generated by
driving a current from an adjacent ferromagnetic material. The basic principle can
be understood by studying Fig. 1.2, which shows the spin dependent density-of-
states (DOS) diagrams of a ferromagnet in contact with a nonmagnetic metal. The
DOS at the Fermi energy for a ferromagnet is different for spin up and spin down,
as shown in the schematic, where spin up has been assumed to be the majority
carrier. In the absence of any current flow between the two metals [Fig. 1.2(a)], the
system is in equilibrium and therefore all the electrochemical potentials (ECP) are
aligned at ?0, the average ECP. In this case the DOS of the nonmagnetic metal is
equivalent for spin up and spin down. By applying a potential difference between
the two metals (by means of a battery, for example), a current can be injected
from the nonmagnetic metal into the ferromagnet [Fig. 1.2(b)]. This means that
electrons will be injected in the opposite direction, that is, from the ferromagnet
(F) into the nonmagnetic metal (N). Since the DOS at the Fermi energy in F is larger
for electrons with spin up than for electrons with spin down, most of the electrons
injected into N will have spin up. This leads to spin accumulation in N as shown
in [Fig. 1.2(b)]. To preserve charge neutrality in each of the metals, an increase in
the number of electrons with spin up is accompanied by a decrease in the number
of electrons with spin down. Therefore charge transport across the F-N interface
is accompanied by spin transport. This produces the nonequilibrium configuration
8
0011
000
000
000
000
111
111
111
111
0000
0000
0000
0000
1111
1111
1111
1111
0000
0000
0000
0000
0000
1111
1111
1111
1111
1111
000
000
000
000
111
111
111
111
0000
0000
0000
0000
1111
1111
1111
1111
000
000
000
000
111
111
111
111
000
000
000
111
111
111
N
? 0
E E
Ferromagnet (F) Nonmagnetic metal (N)
NN N N
? 0
(a)
(b)
E
Nonmagnetic metal (N)Ferromagnet (F)
N NN
E
Figure 1.2: Spin dependent density-of-states diagrams of a ferromagnet placed in
contact with a nonmagnetic metal when (a) no current is being injected, and (b) a
current is driven between the two metals. The dashed line represents the average
electrochemical potential ?0.
of the ECP?s shown in [Fig. 1.2(b)], which, for simplicity, have been assumed to be
continuous at the interface.
Figure 1.3 gives a more complete picture of the charge-spin coupling in a F-N
structure by showing the spin dependent electrochemical potential as a function of
position, with the interface being atx = 0. An electric current is applied between the
nonmagnetic metal and the ferromagnet so that electrons are injected from F into N.
The dashed lines at both sides of the interface represent the electrical potential, with
the slope of the line being proportional to the resistivity of each metal. As explained
above, most of the electrons injected into N have spin up. Therefore ?N? increases
9
,
?N?F
??
NF
?
0 x
Figure 1.3: Position dependence of the electrochemical potential ? for both spin
channels near a F-N interface in the presence of electric current.
with respect to its equilibrium value. This has the following consequences: first,
due to charge neutrality, ?N? has to decrease so that the total DOS at any position
in N is uniform. Since in N the DOS is spin independent, the size of the increase
in ?N? is equivalent to the size of the decrease in ?N? . Second, ?F? has to increase
to avoid back-flow of electrons with spin up from N into F. And third, due to the
increase in ?F? , ?F? should decrease, again to preserve charge neutrality. However,
since the DOS of F is different for spin up and spin down, a large change in ?F? is
necessary to compensate for a small change in ?F? . This produces the asymmetry in
the ECP?s in F shown in Fig. 1.3. This spin-charge coupling is also responsible for
the appearance of the discontinuity in the electrical potential at the F-N interface,
which implies the existence of an additional spin-related voltage at the interface
??/e. In the case of transparent interfaces shown here, the spin dependent ECP?s
are continuous. As the magnitude of the current increases, so does the deviation
in the ECP?s from the zero-current value, which increases the size of ??. The fact
10
that the spin relaxation length ? is longer in N than in F is also clear from Fig. 1.3.
A conventional model for spin injection across the ferromagnet/nonmagnetic
metal (F/N) interfaces [24, 25, 26] is provided by noting that, in each metal, the
spin-flip scattering is typically much weaker than the momentum scattering. This
leads to a mean free path (MFP) l which is much shorter than the spin diffu-
sion length (SDL) ??the characteristic scale for the decay of spin accumulation at
each side of the interface. Within the two-current model one can then define local
spin-resolved electrochemical potentials ??, ? =?,? for carriers with majority and
minority spin (with magnetic moment parallel and antiparallel to the magnetiza-
tion M in a ferromagnet). In the steady state an electrical current driven across a
F/N junction will lead to spin accumulation proportional to (?? ???) which is the
balance between spins added by the magnetization current and spins removed by
spin relaxation. In the absence of interfacial spin-flip scattering, the spin-resolved
current I? is conserved across the interface [25, 26], and the contact resistance for
each spin can be expressed as
R? = e(?F? ??N? )/I?, (1.1)
where the indices F,N label the quantities in the corresponding region at each side
of the contact and e is the proton charge. R? negationslash= R? can be inferred from the effect of
exchange splitting in the F region, leading to spin-dependent Fermi wave vectors,
transmission coefficients, and density of states. Spin accumulation in the N region
can act as a source of spin electromotive force which produces a voltageV ? (?????)
measurable by adding another ferromagnet [27].
11
N21N
B2B1
T
L
N
T (b)
F
(a)
F F
F2
2
1
?
2
Co
L1
1
xCu
Figure 1.4: (a) Sample geometry. Shaded regions represent two Co ferromagnets
separated by a distance L, while the solid black region represents the Cu line. Ti
and Bi are nonmagnetic measurement leads. (b) ?? for spin up and spin down
electrons in injector (left), normal metal (center), and detector (right) as a function
of position. Detector?? for parallel (solid) and antiparallel (dashed) magnetizations
is shown.
Using different variations of spin valve geometries like the one depicted in
Fig. 1.4(a) in which F1 represents the spin injector and F2 the spin detector, spin
injection and detection measurements can be made. Charge current is driven be-
tween the leads T1 and N1 while the nonlocal voltage VNL is measured between the
leads T2 and N2 which, in the absence of nonequilibrium spin, is an equipotential
region without a charge current flow such that VNL = 0. As compared to local
measurements (current driven between B1 and B2 and voltage measured between
T1 and T2), the nonlocal measurement has been shown to simplify the extraction
of spurious effects (for example, anisotropic magnetoresistance and the Hall effect)
from those intrinsic to spin injection [3, 28]. In Fig. 1.4(b) a spatial profile of ?? is
sketched for a fixed magnetization of F1 (pointing ?) and for both aligned (pointing
?, solid traces in region F2) and anti-aligned (pointing ?, dashed traces in regionF2)
magnetization ofF2. The presence of interfacial resistance leads to the discontinuity
12
of the electrochemical potential ? = (?? +??)/2 at each contact [25, 26, 10]. The
region labelled F1 refers to the upper half of the injector, N refers to the middle
section of the nonmagnetic wire between F1 and F2, and F2 refers to the upper half
of the detector. ? and ? label the electrochemical potentials of the two spin channels
in each region. The value of the average electrochemical potential in region F2 is
the measurable quantity, which is directly related to the magnitude of the residual
spin population at the F2 ?N interface. A measurable difference in the average
electrochemical potential of F2 when its magnetization is reversed can be seen as a
difference between the dashed and solid lines at the far right.
1.3 Magnetic force microscopy
On the other hand, magnetic force microscopy MFM is an extremely sensitive prob-
ing tool for measuring extremely small magnetic field gradients with a resolution of
tens to hundreds of nanometers. State of the art cantilevers which require advanced
micro-engineering have been fabricated by groups in Stanford and IBM [29, 30] and
have been used in a special vertical configuration, low temperature, vacuum, mag-
netic resonance force microscope to measure a single localized spin [31]. In this work
I have attempted to use MFM to measure the magnetic field produced not by one,
but by a few hundred electrons which are diffusively moving along a narrow Cu wire,
and which have been injected from a neighboring ferromagnetic electrode. Besides
requiring large sensitivity and resolution, it is essential to separate the very small
magnetic field produced by a few hundred electrons from the very large (1500 times
larger) magnetic fields produced by both the injection current and the ferromagnetic
13
injector. For this purpose high magnetic field resolution cantilevers were fabricated
and the MFM setup was optimized by using a multi-frequency resonant method.
The essence of the method is to drive the injection current at a frequency ?1, oscil-
late the magnetization of the injector at a frequency ?2, and detect the spin signal
at a frequency ?0 = ?1+?2 in resonance with the cantilever natural frequency. Even
though the required sensitivity could not be achieved with the current MFM and
magnetic cantilevers, the resolution and sensitivity were increased. Furthermore, the
measurement setup was refined to significantly reduce unwanted magnetic signals
from currents and magnetic electrodes, and the validity of the operation and model
of the method were validated. Improvements in electrostatic potential balancing
and reduction of the noise level by increasing the quality factor of the cantilever
should make the measurement possible.
14
Chapter 2
Theory of spin transport and decoherence
2.1 Single spin coherence time, spin dephasing, and spin relaxation times
The spin coherence time ?s of an electron in a solid can be defined as follows. Let
|?t? represent the spin state of an electron at time t. Then ?s is the minimum value
of ? that satisfies
limt?+? 1t
integraldisplay t
0
??t?+?|??t?dt? = 0. (2.1)
This condition just states that there should not be any correlation between |?t? and
|?t+?? for ? ??s. Measuring ?s in a real mesoscopic system is not as easy as it may
seem. First, the sensitivity to measure the spin of a single electron in a mesoscopic
system in a time short compared with the decoherence time has not been achieved.
And second, it is difficult to tag one particular electron and distinguish it from the
rest of the electron reservoir1. To get around these two problems, usually the spin
of an ensemble of electrons is measured. By doing so, however, the electron-electron
interactions that conserve the total spin of the electron system are disregarded 2.
1Quantum dots in semiconductor heterostructures have been used to confine one or a few
electrons, separating them from the electron bath; but doing the same in a metallic system is an
insurmountable task.
2Electron exchange interactions and the term in the electron dipole-dipole interaction which
are of the form ??S1 ???S2 conserve the expectation value of the total spin of the electrons ???S? and
therefore are not accounted for in an ensemble measurement.
15
Therefore the ensemble coherence time ?e is longer than the single spin coherence
time ?s. It is important now to make a distinction between measurements made
in the presence or absence of a magnetic field. In the presence of a magnetic field
there will be two different time scales that characterize the exponential approach
to equilibrium of the average spin of the system. These are T1 (the spin relax-
ation, longitudinal, or spin-lattice time) and T2 (the spin dephasing, transverse, or
decoherence time) which characterize the spin decay in directions parallel and per-
pendicular (respectively) to the external magnetic field [32, 33]. T2 measures only
the irreversible decoherence of the electronic spin while T?2 also includes reversible
dephasing between the spins of different electrons due to inhomogeneities in the
magnetic field. Therefore in general T?2 ?T2, but for conduction electrons T?2 = T2
is a very good approximation [34, 13]. These time scales, which were originally
used in nuclear magnetic resonance but which have been adopted also for electron
spin relaxation, are of course ensemble averages. One of the main objectives of
this research is measuring T?2, which is equivalent to the time ?e discussed above
when dephasing is negligible. The other time scale, T1, is usually much longer than
T2 and measures the rate of thermalization of the spin population rather than the
coherence time. For T1 processes, energy is transferred from the spin system to
the lattice, usually by phonons. In the absence of an external magnetic field, or
for small enough magnetic fields 3, and for isotropic systems, the longitudinal and
transverse relaxation times are equal: T1 = T2, and they both measure the spin
3Small meaning that ?L?c << 1 where ?L is the Larmor precession frequency and ?c is the
correlation time (i.e. the typical time interval between spin changing events).
16
coherence time of an ensemble of electrons [32, 35, 36, 37]. The equality between
the relaxation and dephasing times for weak magnetic fields can be understood by
realizing that if the energy uncertainty introduced by the random fluctuations in
the phase of the electron is larger than the Zeeman splitting between the two spin
orientations, then the energy splitting is irrelevant for the dynamics, and the system
is effectively isotropic. Therefore dephasing will affect equally the longitudinal and
transverse components of the spin. Since the Larmor frequencies corresponding to
the magnetic fields applied during the measurements were small compared with the
inverse of the correlation time3, ?L ? 1/?c, and since the material used was copper,
a metal with an isotropic cubic crystal structure, then it is valid to assume T1 = T2,
so throughout the rest of this thesis both the spin relaxation and spin dephasing
times will be referred to with the symbol ?.
2.2 Spin relaxation mechanisms
The relevant sources of spin relaxation in metals are magnetic impurity scatter-
ing, the Elliot-Yafet, and the D?yakonov-Perel? mechanisms. Magnetic impurity
scattering comes from the exchange interaction between the conduction electrons
and a localized magnetic impurity. The Elliot-Yafet mechanism [5, 38] is based on
the fact that in real crystals Bloch states are not eigenstates of spin, and so any
non-magnetic (spin-independent) scattering leads to spin mixing and therefore spin
relaxation. The D?yakonov-Perel? mechanism applies only to crystals that lack in-
version symmetry. In such systems electrons in the same momentum state can have
different energies for spin up and spin down. This can be understood by assuming
17
the reference frame of the electron, in which an effective magnetic field is produced
by the moving charged lattice sites. Since the lattice lacks inversion symmetry, elec-
trons moving in opposite directions will see magnetic fields of different magnitude
so the average field seen by a randomly scattering electron is non-zero. For copper,
which has a cubic lattice, the D?yakonov-Perel? mechanism is irrelevant and the only
sources of relaxation are magnetic impurity scattering and Elliot-Yafet. These will
be treated in more detail in the following sections. Other scattering mechanisms
such as electron spin-spin and electron-nuclear spin give very long relaxation times
in metals (> ?s) [39] when compared with the Elliot-Yafet and magnetic impurity
scattering, and will be ignored.
2.2.1 Elliot-Yafet
Conduction electron spins can relax through ordinary momentum scattering with
phonons and impurities since eigenstates of spin are not eigenstates of the complete
Hamiltonian of the system. The spin-orbit interaction introduces a term in the
Hamiltonian of the form
VSO = ?h4m2c2?U ? ?p? ??, (2.2)
whereU is thespin-independent periodicpotential ofthe lattice, ?pis themomentum,
and ?? are the Pauli matrices. For metals with inversion symmetry the Bloch states
for electrons with spin up and spin down with crystal momentum k and band index
n are given by
18
?k,n,?(r) =
bracketleftBig
ak,n(r)|??+bk,n(r)|??
bracketrightBig
eik?r
?k,n,?(r) =
bracketleftbigg
a??k,n(r)|???b??k,n(r)|??
bracketrightbigg
eik?r. (2.3)
Since the spin-orbit coupling term is much smaller than the kinetic and potential
terms of the Hamiltonian, it can be treated perturbatively. The value of the coef-
ficient bk,n(r) is close to zero whereas that of ak,n(r) is close to one. Hence, even
though the eigenstates of the full Hamiltonian have mixed spin, it still makes sense
to talk about spin up and down, as those states with a majority of spin up or down.
By itself, spin-orbit coupling does not cause spin relaxation. However, the presence
of any kind of momentum scattering does. Since the non-diagonal elements of the
scattering matrix with different spin indexes are non-zero, transitions between states
with different spin are allowed and can occur every time there is some scattering
event. This means that mechanisms that produce momentum relaxation such as
phonons, impurities, and boundaries, will also produce spin relaxation. Elliot and
Yafet showed that for negligible heavy impurity scattering the ratio of the momen-
tum and spin relaxation times is given by a constant
ap = ?p/?. (2.4)
At high temperatures phonons are the main source of scattering. The tempera-
ture dependence of the spin relaxation time then follows the temperature dependence
of the mean free time 1/T, so the spin relaxation time has the same temperature
19
dependence as the conductivity. In the absence of any other scattering mechanism
besides phonons, the spin relaxation time would also be proportional to the mean
free time at low temperatures. Since in metals at low temperatures the resistivity
behaves according to Bloch?s law, ? ? T5, the spin relaxation time should have a
T?5 dependence. However, spin relaxation at low temperatures is caused mainly by
the temperature independent impurity and boundary scattering. Therefore the spin
relaxation time becomes temperature independent at low temperatures, making the
observation of the T?5 behavior impossible up to now.
2.2.2 Magnetic impurity scattering
For dilute concentrations of magnetic impurities ina nonmagnetic host, the exchange
interaction between the localized spin and the conduction electron spins creates
an additional mechanism for spin relaxation and momentum scattering. When a
conduction electron is scattered off a magnetic impurity, the spin of the electron is
flipped, while the localized magnetic moment changes in order to compensate for the
gained or lost magnetic moment of the electron. Since momentum is also affected
in the magnetic scattering, this process contributes to the resistivity (Kondo effect)
[40]. The temperature dependance of the Kondo resistivity is given by [41]
?K = c?M[1?bln(T/TK)], (2.5)
where ?M is a measure of the strength of the exchange scattering, c is the magnetic
impurity concentration, b depends on the ratio of the exchange and Fermi ener-
gies and on the crystal structure, and TK is the Kondo temperature, which gives
20
the temperature where the Kondo resistivity becomes comparable to the phonon
induced resistivity. The temperature dependence of the Kondo resistivity is rather
different from the phonon resistivity, since it increases with decreasing temperature.
The competing effects of the two resistivity mechanisms gives a minimum in the
resistivity at a temperature which depends on the relative magnitude of phonon
and magnetic impurity scattering. For Fe impurities in thin Cu films the Kondo
temperature is of the order of 15oK [42], and the Kondo effect is suppressed as the
thickness of the film is decreased.
However, not every magnetic impurity causes Kondo behavior. For example
Ni impurities in Cu do not, and Co impurities in Cu create a not very well defined
magnetic moment which gives a quasi-Kondo temperature of 500oK [43]-1000oK [44].
However, Co atoms in the surface of Cu are considered to be a Kondo system [43]
with a Kondo temperature of about 23oK. Well above the Kondo temperature the
main source of resistivity, and of spin relaxation, is due to phonons. However at
4.2oK the main source of resistivity and spin relaxation is a combination of magnetic,
nonmagnetic impurity scattering, and boundary scattering4. The relative magnitude
of these effects depends on the magnetic and nonmagnetic impurity concentration,
the spin orbit-coupling, and the grain boundary scattering. By using high purity
nonmagnetic metals (5 ppm or less impurities), it is possible to significantly reduce
the magnetic impurity scattering so at 4.2oK the main source of resistivity and spin
relaxation is boundary scattering [42].
4Specially important in films with grain structure.
21
Chapter 3
Spin Transport
In the previous chapter characteristic time scales foran ensemble of spins, such as the
relaxation and the dephasing time, were defined, and the origins of spin relaxation
and dephasing in metals with inversion symmetry was discussed. In this chapter I
will discuss how to measure the spin relaxation time in metals by providing the basic
theory for spin transport in one-dimension and by applying it to spin-valve devices
with different geometries and different types of contacts. I will also discuss the effect
of interfacial spin-flip scattering and small leakage currents. However, before going
on to present the theory of spin transport, I will, as a motivation, describe and
briefly explain the operation of a spin-valve device.
A typical spin-valve device is shown schematically in Fig. 3.1. The gray areas
represent magnetic contacts, the black regions are normal metal wires, and the white
regions are leads to inject or sink current and to measure voltages. The magnetic
contacts serve as a source of spin-polarized electrons and also as a way of detecting
the remaining magnetization of the injected spin-polarized electrons at a distance
L1 +L2 from the injection point. In normal operation, a current is injected from
one of the leads, say B1, and the ground of the system is connected to lead BN.
22
F1 F2
I
V
000
000
000
000
111
111
111
111
000
000
000
000
111
111
111
111
B1 BN
T2
L 2N
T
L 1
0
0
1
1
0
0
1
1
Figure 3.1: Schematic of a spin valve device showing magnetic contacts (gray),
normal metal wires (black), and normal metal leads (white) for current transport
and voltage measurement.
Due to the exchange splitting in ferromagnetic materials, spin up and spin down
conduction electrons have different number densities and Fermi velocities, so the
current injected from the magnetic contact labelled F1 into the normal metal wire
is spin-polarized. As the electrons travel along the normal wire between F1 and the
cross, and as they diffuse from the cross to F2, part of the spin-polarization is lost
due to spin relaxation. When they reach F2 the spin-polarization of the electrons in
the normal metal wire is different from that of the electrons in the magnetic contact
F2, so an electrochemical potential difference appears between the two metals. This
potential difference can be observed by connecting leads at TN and T2. Depending
on the relative alignment of the magnetization ofF1 andF2 (injector and detector)1,
1Which can be changed by applying an external magnetic field to selectively flip the magneti-
zation of the injector or detector due to their different aspect ratios.
23
the measured potential is different. This measurement provides some information
on the degree of injected spin-polarization and on the spin relaxation time. Details
of the operation of the spin-valve will be addressed later in this section. Now the
relevant theory for diffusive spin transport will be presented.
3.1 Spin transport: diffusion equation
Intheabsence ofanymomentum scattering mechanism (phonons, impurities, bound-
aries, other electrons, etc.), electrons in a crystal lattice are in Bloch states, eigen-
states of the periodic lattice potential, and are able to transport charge without any
resistance. In such a case transport is ballistic in any length scale. However in real
systems at non-zero temperatures, in particular in mesoscopic systems where the
length scales are of the order of tens or hundreds of nanometers and where single
metallic crystals are difficult to grow, impurities, phonons, and boundaries prove
to be effective momentum scattering mechanisms, so that the average time that an
electron can travel without any scattering, the mean free time?c, is of the order of 10
fs2. For a fermi velocity of 106m/s this corresponds to a mean free path of the order
of 10 nm. Therefore, from the point of view of the mean free path, a mesoscopic wire
with a cross section of 50 nm by 100 nm is effectively three dimensional. Transport
in metallic mesoscopic wires over distances of the order of a micron (typically much
larger than the mean free path) is diffusive, which means that in a time t, electrons
travel an average distance x given by
2Mesoscopic devices with long mean free times can be prepared, but for the particular devices
fabricated for this research, and for typical mesoscopic devices found in the literature, mean free
times of the order of 10 fs are typically observed.
24
x = ?Dt, (3.1)
after being scattered many times by phonons, impurities, and boundaries, where D
is the diffusion constant given by
D = 13V2F?c, (3.2)
with VF the Fermi velocity of electrons in the metal. In the Drude model the mean
free time is related to the conductivity? of the metal by? = ne2?c/m? wherenis the
free electron density, e is the electron charge, and m? the effective electronic mass.
An alternative, more useful form of equation (3.2) is given by Einstein?s relation [41]
D = ?e2N, (3.3)
where N is the density of states at the Fermi energy.
Typical spin relaxation times measured using methods like CESR (conduction
electron spin resonance) [45, 46] or spin injection [3] are between tens of picoseconds
and nanoseconds, although for high purity sodium at 10 K values as large as 1 ?s
have been measured [47]. Since the spin relaxation time is typically much larger
than the mean free time, it is possible to define local distinct electron densities for
spin up n? and spin down n?. Even though there is no special direction that can
define up and down, it is possible to choose any direction as the quantization axis.
Starting from the Boltzmann transport equation and using the approximation of a
25
linear transport regime at zero temperature, it can be shown [10] that the current
density j? for spin ? (? =?,?) is given by
j? = ??E?eD???n?, (3.4)
where ?? is the conductivity, E the electric field, D? the diffusion constant, and
?n? the deviation from equilibrium of the local carrier density. The first term
corresponds to Ohm?s law while the second one is due to diffusion from high to low
electron concentration. By definition, the density of states at the Fermi energy N?
for each spin channel is N? = ?n?/??? where ?? is the chemical potential. Replacing
this, together with equation (3.3) into equation (3.4) gives
j? = ???e ???, (3.5)
where the electrochemical potential ?? is defined as
?? = ?? +e?, (3.6)
where ? is the electric potential which produces the electric field E. Due to relax-
ation, spin is not conserved. Therefore the continuity equation is given by
??j? +e?n?/?? = 0, (3.7)
where ?? is the time for an electron to scatter out of state ?. From equation (3.7)
it is possible to obtain the charge and spin continuity equations
26
??(j? +j?) = 0 (3.8)
??(j? ?j?) = ?e(?n?/?? ??n?/??). (3.9)
By replacing j? from equation (3.5) into equations (3.8) and (3.9) and using the
detailed balance principle N?/?? = N?/?? [48, 49], which states that the scattering
rates for spin ? and spin ? are equivalent at equilibrium, we obtain [26]3
?2(???? +????) = 0 (3.10)
?2(?? ???) = ??2(?? ???), (3.11)
with the spin diffusion length ? related to the spin relaxation time
? = 2
parenleftBigg1
?? +
1
??
parenrightBigg?1
, (3.12)
and the diffusion constant
D = (N? +N?)D?D?N
?D? +N?D?
, (3.13)
in the usual way, i.e. ? = ?D?. In the case of normal metals N? = N? = N/2,
D? = D? = D, ?? = ?? = ?/2, and ?? = ?? = ?.
3Although the derivation of Takahashi [26] was followed, it agrees with Valet and Fert [10] and
Jedema [49] except for the definition of ?, which was incorrect in [49].
27
3.1.1 Spin transport in one dimension
For spin diffusion in one dimension (say in the x direction) the most general solution
to equations (3.10) and (3.11) is
?? = a+bx? c?
?
e?x/? ? d?
?
ex/?, (3.14)
where the positive sign is for ? =? and the negative sign for ? =?, and a, b, c, and
d are still unspecified coefficients. This solution is valid for both magnetic and non-
magnetic metallic regions. The first two terms on the right hand side of equation
(3.14) are the same for both spin channels. They represent a reference potential
energy plus a linear (in x) potential energy coming from the electric field that drives
the current. In the absence of charge current the second term vanishes. The last two
terms on the right hand side of equation (3.14) have different sign and magnitude
for the different spin channels and they show the exponential decrease or increase in
the electrochemical potential as a function of position. In order to find the values of
the coefficients it is necessary to specify the particular geometry of the device and
a set of boundary conditions. This will be done in the following sections.
3.1.2 Interfacial spin transport
The previous sections presented the formalism for describing diffusive spin transport
in metals. However, no reference was made to transport across interfaces, which is
critical for the injection and detection of spin, as explained at the beginning of
this chapter (see Fig. 3.1). Therefore the issue of interfacial spin transport will be
28
addressed in this section.
A complete understanding of the microscopic details of spin transport across
interfaces is lacking. Even though specific details of the interface seem to be relevant,
first principle calculations are incomplete, and it is necessary to use phenomenolog-
ical parameters and make assumptions with no a-priori justification, which can be
validated only if the predictions agree with the measurements. Here the same phe-
nomenological approach will be taken. It is usually assumed [10, 22, 50, 51] that
the spin-flip scattering at the interface is irrelevant. This assumption will be used
for the most part of this chapter but it will be relaxed in section 3.5.1 where the
effects of interface spin-flip scattering will be studied. If this assumption holds,
the spin-dependent current must be conserved at each interface. This defines the
first boundary condition for the electrochemical potential. In addition, the inter-
face between two metals can be either transparent or resistive (non-transparent).
Transparent contacts are defined as those in which the electrochemical potential is
continuous across the interface. Resistive contacts, on the other hand, are those
in which the difference in electrochemical potential between the two sides of the
interface is equal to the product of the spin-dependent resistance R? and the spin-
dependent current j?A that crosses the interface:
?N? ??F? = R?j?A, (3.15)
where A is the area of the contact, and F and N label ferromagnetic and normal
29
1/R
1/R
FN
Figure 3.2: Schematic of the transport across a nonmagnetic-ferromagnet interface
in the case of conserved spin. The two transport channels (spin ? and spin ?) and
their associated interface resistances are shown.
regions of the device.4 This sets the second boundary condition for the electrochem-
ical potential. A schematic of the spin conserved transport at the interface is shown
in Fig. 3.2 for aligned injector and detector. The dots on the left represent the non-
magnetic metal while the ones on the right represent the ferromagnetic metal. The
two possible transport channels (spin ? and spin ?) are shown, together with their
associated interface resistance. Different interface resistances for spin up and spin
down are due to the exchange splitting in the ferromagnetic region, which leads to
spin-dependent Fermi wave vectors, transmission coefficients, and density of states.
The remaining boundary condition is that the electrochemical potential be finite
as x ? ??. This three boundary conditions, together with information on the
geometry of the spin-valve, are enough to find the coefficients that specify the exact
functional form of the electrochemical potential in every region of the device.
4Only F ?N contacts will be studied since they are essential for spin injection and detection
(see Fig. 3.1).
30
1F
1N
1
N T
2T
2L
1
L
2
N 2
F
BNB 2
T
B
3
1
y4 6
x
5
2
1
y??y?
Figure 3.3: Schematic of a spin valve device with magnetic contacts (gray), normal
metalwires (black), andnormalmetalleads (white) forcurrent transport andvoltage
measurement, in the cross geometry configuration.
3.2 Spin valve structures
Several geometries of spin-valve devices were fabricated and different measurement
configurations were used. They can be divided into two different kinds: cross ge-
ometry and nonlocal geometry. In the following sections I will solve the equations
for the electrochemical potential for the two different configurations, and calculate
relevant quantities such as the spin-related resistance and the contact resistance,
which are the quantities that can be observed experimentally.
3.2.1 Cross geometry
The cross geometry configuration is shown in Fig. 3.3. The black regions are made
of a nonmagnetic metal, the white regions are non-magnetic leads for probing the
system, and the gray areas are the ferromagnetic contacts. F1 and F2 label each
31
of the two magnetic regions, L1 and L2 give the distances between the cross and
each of the magnetic contacts, while the remaining capital letters label leads that
are available experimentally. The two ferromagnetic contacts are elongated in the
vertical direction (along the y axis) to facilitate alignment of the magnetization in
this direction. Therefore the vertical axis is called the easy geometrical axis. Addi-
tionally, the ferromagnetic contacts have different aspect ratios to ensure different
coercive fields, and hence allow independent rotation of the magnetization of each
contact with an applied external field. The active part of the device, in which spin
injection and detection takes place, is composed of the black and gray regions, which
can be divided into six different sections numbered in Fig. 3.3. Region 1 is the lower
half of F1, regions 2-5 are the different arms of the cross, with regions 2 and 3 ex-
tending from the middle of the cross to the vertical black lines at the centers of F1
and F2 respectively, and region 6 is the upper half of F2. The dashed black lines
with arrow heads indicate the coordinate systems that will be used. Region 1 is
along ?y?, region 2 along ?x, region 3 along x, region 4 along y, region 5 along ?y,
and region 6 along y??.
Spin injection and detection experiments are performed by injecting current
into lead B1, grounding lead BN so that all the charge current is drained through
this lead, and measuring the potential difference between leads TN and T2, between
which no charge current circulates 5. In such a configuration, the equations for the
5This is not exactly true since as the current flows throughthe cross, a measurableohmic voltage
appears between leads TN and T2. However, this voltage is independent of the magnetization
direction of the magnetic contacts and can therefore be distinguished from the spin signal.
32
electrochemical potentials in the six different regions described above are
?1?(y?) = a1 ? je?
F
y? ? d1?? exp y??
F
?2?(x) = a2 ? je?
N
x? c2?
N
exp ?x? ? d2?
N
exp x?
?3?(x) = a3 ? c3?
N
exp ?x? ? d3?
N
exp x?
?4?(y) = ? c4?
N
exp ?y?
?5?(y) = a5 + je?
N
y? d5?
N
exp y?
?6?(y??) = a6 ? c6?? exp ?y???
F
, (3.16)
with?F = ??+?? and?N the conductivities of the ferromagnet (F) and normal metal
(N), ?F and ? the spin relaxation lengths in (F) and (N), j the current density, and
where the positive (negative) sign is for ? =? (? =?). Charge current conservation
has been used to solve for the coefficient of the linear term in the electrochemical
potential, assuming equal cross sectional areas for all the regions6. Since the poten-
tial is specified up to a constant, the constant term in the electrochemical potential
for the lead labelled TN (region 4 in Fig. 3.3) was set to zero. The condition of
finite electrochemical potentials at y? ? ??, y ? ??, and y?? ? ?, simplified
the original equations by removing the exponentially divergent terms. Conservation
of spin current, which has not been used yet, gives the following set of boundary
conditions, one for each of the three points at which different regions meet7:
6This assumption can be relaxed and the case of different cross sectional areas for F1, F2, and
N can be obtained by multiplying the second term in the first one of equations (3.16) by AN/AF,
the ratio of the cross sections of the normal metal wire and the ferromagnetic contact F1.
7Again for equal cross sectional areas.
33
???1?(0)?? +??2?(?L1)?N/2 = 0
???2?(0) +??3?(0) +??4?(0)???5?(0) = 0
???3?(L2)?N/2+??6?(0)?? = 0, (3.17)
where ?? is the derivative of the electrochemical potential. The most relevant coeffi-
cient in equations (3.16) is a6 which gives the potential at lead T2 far away from the
active region of the device. The measured signal, as explained above, is the voltage
V between leads TN and T2 which is given by
eV = ?4(y ??)??6(y?? ??) = ?a6. (3.18)
Cross geometry spin valve with transparent contacts
In the case of transparent contacts, the additional boundary conditions are the
continuity of the electrochemical potentials for both channels:
?1?(0)??2?(?L1) = 0
?2?(0) = ?3?(0) = ?4?(0) = ?5?(0)
?3?(L2)??6?(0) = 0. (3.19)
Equations (3.17) and (3.19) are valid for the case of aligned injector F1 and detector
F2. However, if they are anti-aligned, the direction of the majority spin for F1 will
be the same as the direction for the minority spin of F2. This means that to obtain
34
0 0.2 0.4 0.6 0.8 1
L (?m)
0
0.25
0.5
0.75
1
1.25
1.5
R
(m
?)
1 ? 2 =?
Figure 3.4: Resistance given by equation (3.20) as a function of separationLbetween
magnetic contacts for ?1?2 =?? (aligned magnetizations of F1 and F2).
the correct boundary conditions in the case of anti-aligned magnetic contacts, the
indexes ? and ? in equations (3.17) and (3.19) have to be reversed for either F1 or
F2. The solution to equations (3.16) with boundary (matching) conditions (3.17)
and (3.19) in the case of L1 = L2 = L/2 gives an observable voltage V (3.18) which
can be normalized by the injected current to obtain the spin related resistance
R = ? ?
2
F?
?NA(1 +?(1??2F))[1??(1??2F) + exp(L/?)(1 +?(1??2F))], (3.20)
where ?F = ?? +??, ?F = (?? ???)/?F, ? = ??F/(?F?N), A is the cross sectional
area of the wires 8, and the positive (negative) sign is for aligned (anti-aligned)
magnetizations of F1 and F2. Figure 3.4 shows the behavior of the resistance given
by equation (3.20) as a function ofLfor typical values of the relevant parameters for
8Again in the case of regions with equal cross sectional area.
35
+x
0 0.05 0.1 0.15 0.2 0.25 0.3
Position (?m)
? 1
0
1
2
3
?(
?eV
)
y???x
? 1 ? 2 =
?
?
? 1 ? 2 =
Figure 3.5: Electrochemical potential for both spin channels and both aligned (solid
line) and anti-aligned (dashed line) magnetizations of F1 and F2.
Cu. It shows that the spin-related resistance decays with L, so that measurements
of R for different values of L can give information about the relaxation length of
electronic spin in the normal metal.
The behavior of the electrochemical potential as a function of position for fixed
L1 = L2 = L/2 =0.2?m is shown in Fig. 3.5 for both spin channels and both aligned
(?1?2 =??, solid line) and anti-aligned (?1?2 =??, dashed line) magnetizations of F1
and F2. The solid and dashed lines coincide throughout F1 and N but they differ in
F2. The axes at the top delimit regions 2 (-0.2?m<x<0), 3 (0<x<0.2?m), and
6 (y?? > 0.2 ?m) as defined in Fig. 3.3. The difference in electrochemical potentials
(ECP?s) between the two configurations ?? and ?? in the region in F2 where the
spin has relaxed completely is the measurable signal, equal to twice the potential V
36
+x
? 0.2 ? 0.1 0 0.1 0.2
Position (?m)
? 7.5
? 5
? 2.5
0
2.5
5
7.5
? ?
??
?
(?
eV
)
y???y? ?x
? 1 ? 2 =
? 1 ? 2 =
Figure 3.6: Spin component of the electrochemical potential for aligned (solid line)
and anti-aligned (dashed line) magnetizations of F1 and F2.
[see equation (3.18)] in units of eV. The ohmic potential drop in region 2 (see Fig.
3.3) can be seen as a line with large negative slope for both spin channels. The slow
change of the ECP in region 3 and the fast change in region 4 is due to the much
larger relaxation length in nonmagnetic metals than in ferromagnetic ones.
The ECP?s shown in Fig. 3.5 contain both spin and charge effects. The spin
effects can be isolated by calculating the spin electrochemical potential (SECP)
which is the difference in the ECP of both spin channels, ?? ??? as shown in Fig.
3.6 for both aligned (solid line) and anti-aligned (dashed line) magnetizations of F1
and F2. The axes at the top delimit regions 1 (y? <-0.2 ?m), 2 (-0.2 ?m< x <
0),3 (0 < x < 0.2 ?m), and 6 (y?? > 0.2 ?m) defined in Fig. 3.3. Far inside F1,
the component of the electrochemical potential related to spin (SECP) is zero, since
37
the spin population is in equilibrium. However, since spin is being injected into the
normal metal, there is a large non-equilibrium spin population at theF1?N interface
which manifests as a large SECP. Back-diffusion of the electrons into the injector
also increases the SECP for a short distance of the order of the spin relaxation length
in the magnetic material ?F. As electrons are transported to the cross (Fig. 3.3)
by the current flow, spin relaxation decreases the degree of spin polarization and
therefore the SECP. At the cross the charge current flows in a different direction and
the spin polarization can only be transported to region 3 by the diffusive motion
of electrons. Hence the SECP decreases at a smaller rate. Finally, the residual
non-equilibrium spin population at the F2 ?N interface diffuses a distance ?F into
F2 and the SECP decays to zero.
Cross geometry spin valve with non-transparent contacts
In the case of non-transparent contacts the electrochemical potential is no longer
continuous at theF?N interfaces so equations (3.19) have to be modified to include
the spin-dependent contact resistances R1? andR2? (for contactsF1?N andF2?N
defined in equation [3.15]) as follows:
?1?(0)??2?(?L1) = ???2?(?L1)Ac1R1??N/2
?2?(0) = ?3?(0) = ?4?(0) = ?5?(0)
?3?(L2)??6?(0) = ???3?(L2)Ac2R2??N/2, (3.21)
where Ac1 and Ac2 are the contact areas for each of the two F ?N contacts, and ??
38
is the derivative of the electrochemical potential. In this case the complete solution
is long and not very enlightening so its transcription has been avoided. However, in
the limit of large R1? and R2? compared to the resistance of each material (F, N)
over its spin relaxation length, the measurable spin-related resistance is found to be
R = ? ?P1P2A?
N[exp(L/?) + cosh((L1 ?L2)/?)]
, (3.22)
whereL = L1+L2 is the totalseparation between magnetic contacts,A = Ac1 = Ac2,
the positive (negative) sign is for aligned (anti-aligned) magnetizations of F1 and
F2, and
Pi = Ri? ?Ri?R
i? +Ri?
, (3.23)
for i = 1,2. Furthermore, P1 can be interpreted as the spin current polarization
injected into the normal metal defined as
P1 = j1? ?j1?j
1? +j1?
, (3.24)
where j1? and j1? are the current densities for spin up and spin down at the F1-N
interface. The equivalence between equations (3.23) fori = 1 and (3.24) is obtained
by explicitly calculating the spin dependent current densities j1? using the same set
of equations for the electrochemical potential and boundary conditions that led to
equation (3.22). On the other hand, P2 cannot be interpreted directly as the spin
current polarization at the detector: since no charge current flows across the N-F2
interface, the spin current polarization at the detector would be infinite. However,
39
it will be shown in chapter 5 that P2 contains important physical information of
the ratio of spin polarized transport to total transport across the detector inter-
face. Therefore the Pi are not simply a convenient ratio of the resistances, but also
provide information about the degree of spin-polarized transport across each of the
interfaces. The degree of spin polarization is set then, not by the interplay of the
resistivity and the spin relaxation lengths of the metals, but instead by the values
of the spin-dependent contact resistances. This means that choosing an appropriate
magnetic material, which defines the exchange splitting and therefore the density of
states at the Fermi energy, and choosing the resistive barrier between the F and N
materials, which changes the tunnelling probabilities, is essential for maximizing spin
injection. Figures 3.7 and 3.8 show respectively the electrochemical potential and
the spin electrochemical potential for aligned (solid line) and anti-aligned (dashed
line) magnetizations of F1 and F2, for spin current polarizations P1 = P2 = P of (a)
5%, (b) 10%, and (c) 20% for a constant contact resistance of 5 ?. The spin-related
resistance (proportional to the difference between the solid and dashed lines at the
right of each of Figs. 3.7), as well as the magnitude of the chemical potentials,
clearly increase with P. However, more information can be extracted by looking at
the SECP in Fig. 3.8. The spin signal again increases in magnitude with increasing
P, but there are some additional important features to be noticed. In the extreme
case of P=0 (not shown here), equal number of electrons with spin up and spin
down cross the interface. Since F1 has, in equilibrium, more electrons with spin up
than spin down, the ratio between spin up and spin down electrons is increased,
40
?Position (  m)
(   eV)
p=5%
?
0 0.05 0.1 0.15 0.2 0.25 0.3
-1
0
1
2
3
4
0 0.05 0.1 0.15 0.2 0.25 0.3
-2
0
2
4
6
0 0.05 0.1 0.15 0.2 0.25 0.3
-1
0
1
2
3
p=10%
p=20%
?x +x y??
(a)
(b)
(c)
? 1 ? =
?
2
? 2 =1
? 1 ? 2
? 1
=
2 =?
1 ? 2 =??
?
?
?
?
?
?
? 1 ? 2 =
Figure 3.7: Electrochemical potential for both spin channels and both aligned (solid
line) and anti-aligned (dashed line) magnetizations of F1 and F2 for spin current
polarizations P1 = P2 = P of (a) 5%, (b) 10%, and (c) 20% for a contact resistance
of 5 ?.
41
producing an enhancement in the SECP (directly related to the magnetization) in
F1 close to the interface. This effect is seen in Fig. 3.8(a). The value of P is only
5% so only a few more electrons with spin up than electrons with spin down cross
the interface, effectively increasing the ratio of spin up to spin down electrons in F1,
producing the ?spike? at x =-0.2?m. In (b) this is less evident since P is higher
so more electrons with spin up are leaving F1, and in (c), for P =20%, the effect
has now reversed, showing that, compared with the number of electrons with spin
down, the number of electrons with spin up that leave F1 is so large that the ratio
of electrons with spin up to electrons with spin down left in F1 decreases below
the equilibrium value, giving a dip in the SECP at the interface. This reversal oc-
curs when the spin current polarization at the injector interface P is equal or larger
than the spin current polarization of the magnetic material PM. For the simulations
shown in Figs. 3.7 and 3.8 the valuePM =16.7% has been used9. Another important
observation is that for P=5% the injected spin polarization is small so not many
electrons can pile up in the right hand side of the injector interface (x=-0.2?m+?),
or equivalently, the magnetization diffuses from the interface towards the right at
a faster rate than the maximum injection rate of F1. This is different for P=10%,
where the spin polarized electrons pile up at the interface, acting as an additional
impedance for injection. In this case the rate at which the magnetization diffuses
to the right is smaller than the maximum rate of injection. This is known as spin
bottle-neck.
9Even though Giant Magneto Resistance (GMR) measurements give values for PM as high as
50% for Co [52], much smaller values have also been measured [49].
42
(c)
?
?
(   eV)?
?
-0.2 -0.1 0 0.1 0.2
-10
-5
0
5
10
-0.2 -0.1 0 0.1 0.2
-20
-10
0
10
20
-0.2 -0.1 0 0.1 0.2
-7.5
-5
-2.5
0
2.5
5
7.5
?y? ?x +x y??
Position (  m)?
(a)
p=5%
p=10%
(b)
p=20%
1 ? 2 =
? 1 ? 2 =
? 1 ? 2 =
? 1 ? 2 =
? 1 ? 2 =
? 1 ? 2 =
?
Figure 3.8: Spin electrochemical potential for both aligned (solid line) and anti-
aligned (dashed line) magnetizations of F1 and F2 for spin current polarizations
P1 = P2 = P of (a) 5%, (b) 10%, and (c) 20% for a contact resistance of 5 ?.
43
3.2.2 Nonlocal geometry
The geometry discussed in the previous section has the drawback that voltages
which are not spin-related appear at the measuring leads, and that a large part of
the spin polarized electrons diffuse into the vertical arms of the cross, decreasing the
measured signal. To avoid these problems, the nonlocal configuration shown in Fig.
3.9 is preferred. For this type of geometry the current is injected through lead B1
while leadN1 is grounded so all the current is sinked this way. The voltage difference
between leads T2 and N2 is measured. The ratio of the measured nonlocal voltage
to the injected current is the nonlocal resistance R. The nonlocal character of the
measurements comes from the fact that any effect produced by spin comes from
purely diffusive transport of the magnetization (spin) without any accompanying
charge current, and in the absence of spin effects the measured voltage would be
x
y1
F
N
1
1
T2T
2
N
F
2B
2
B
y?
1
3 4
L
56
2
71
Figure 3.9: Schematic of a spin valve device for nonlocal measurements. Gray
regions represent the magnetic contacts, black ones the normal metal wire, and
white regions the connecting leads.
44
zero. In other words, spin current flows in regions 3, 4, 5, 6, and 7, but no charge
current flows in such regions, so spurious effects like anisotropic magnetoresistance
or Hall effect cannot affect the measurement. The procedure for finding the relevant
equations is very similar to that described in the last section for the cross geometry,
so details will not be mentioned.
For this geometry the equations for the electrochemical potential [see equation
(3.14)] are
?1?(y) = a1 ? Ie?
FAF1
y? d1?? exp y?
F
?2?(x) = a2 + Ie?
NAN
x? d2?
N
exp x?
?3?(x) = a3 ? c3?
N
exp ?x? ? d3?
N
exp x?
?4?(x) = ? c4?
N
exp ?y?
?5?(y?) = a5 ? c5?? exp ?y??
F
?6?(y) = a6 ? c6?? exp ?y?
F
?7?(y?) = a7 ? d7?? exp y??
F
, (3.25)
where charge current conservation has been used to find the coefficients of the linear
terms, I is the injected current, and AF1 and AN are the cross sectional areas of F1
and N. The potential in region 4, far away from the active area of the device, was
chosen as zero so that the measurable nonlocal resistance R is
R = ?5(y
? ??)??4(x??)
Ie =
a5
Ie. (3.26)
45
Since the chemical potential is continuous in regions made of the same material, it is
clear from equations (3.25) that a6 = a1, c6 = d1 a7 = a7, and d7 = c5. For aligned
magnetic contacts the continuity in the spin current at the F1 ?N and F2 ?N
interfaces implies
???1?(0)??AF1 ???2?(0)?N/2AN +??3?(0)?N/2AN = 0
???3?(L)?N/2AN +??5?(0)??AF2 +??4?(L)?N/2AN = 0, (3.27)
where AN, AF1, and AF2 are the cross sectional areas for the normal metal and
ferromagnetic contacts F1 and F2, and ?? is the derivative of the electrochemical
potential. For anti-aligned magnetic contacts the spin index for either regions 1 and
6 or regions 5 and 7 should be reversed.
Nonlocal geometry spin valve with transparent contacts
For transparent contacts the electrochemical potential (ECP) is continuous therefore
the last set of boundary conditions for aligned magnetizations of F1 and F2 is
?1?(0) = ?2?(0) = ?3?(0)
?3?(L) = ?4?(L) = ?5?(0). (3.28)
In the case of anti-aligned magnetizations of F1 and F2 the spin index for either
region 1 or region 5 should be reversed. By applying the conditions (3.27) and
(3.28) to equations (3.25), the nonlocal resistance is found to be
46
R = ?
2RN exp(?L/?)(?FRF1/RN1??2
F
)(?FRF2/RN1??2
F
)
(1 + 2RF1/RN1??2
F
)(1 + 2RF2/RN1??2
F
)?exp(?2L/?), (3.29)
where RF1 = ?F/(2?FAF1), RF2 = ?F/(2?FAF2), RN = ?/(?NAN), and the posi-
tive (negative) sign is for aligned (anti-aligned) magnetizations of F1 and F2. The
factors of 2 in the definition ofRF1 andRF2 come fromthe existence of the additional
regions 6 and 7 (which had been ignored in previous analysis [26]).
Nonlocal geometry spin valve with non-transparent contacts
In the case of resistive contacts the boundary conditions for the ECP are
?1?(0)??3?(0) = ???1?(0)Ac1R1???
?2?(0) = ?3?(0)
?3?(L) = ?4?(L)
?3?(L)??5?(0) = ???5?(L)Ac2R2???, (3.30)
where Ac1 and Ac2 are the junction areas of the contacts F1 ?N and F2 ?N, and
?? is the derivative of the electrochemical potential. In the case of anti-aligned
magnetizations of F1 and F2 the spin index for either region 1 or region 5 should be
reversed. By applying conditions (3.27) and (3.30) to equations (3.25), the nonlocal
resistance is found to be
R = ?
2RN exp(?L/?)( P1Rc1R
N(1?P21 )
+ ?FRF1R
N(1??2F)
)( P2Rc2R
N(1?P22 )
+ ?FRF2R
N(1??2F)
)
(1 + 2Rc1R
N(1?P21 )
+ 2RF1R
N(1??2F)
)(1 + 2Rc2R
N(1?P22 )
+ 2RF2R
N(1??2F)
)?exp(?2L/?), (3.31)
47
where Rcj (j = 1,2) is given by
Rcj = AcjA
Fj( 1Rj? + 1Rj?)
. (3.32)
It can be seen that in the limit of transparent contacts, Rc1 = Rc2 = 0, equation
(3.29) is recovered. In the opposite limit of contact resistance much larger than the
resistance of the ferromagnetic and normal metals over their spin relaxation length,
more precisely when
Rcj >>RFj,RN (3.33)
for j = 1,2, equation (3.31) reduces to
R = ?12P1P2RN exp(?L/?), (3.34)
where the positive (negative) sign is for aligned (anti-aligned) magnetizations of F1
and F2. This equation takes into account the effect of all the electrons that diffuse
from injector to detector in time t < ?. However, it is also useful to consider the
effect of only those electrons which diffuse from injector to detector in exactly a
time t. This is given by
RD(t) = ?12P1P2RN exp(?Dt/?2). (3.35)
From the theory of diffusive transport (or equivalently by solving the symmetric
random walk problem) it can be shown that the probability for an electron to diffuse
48
a distance L in a time between t and t+dt in a system with diffusion coefficient D
is given by
P(t)dt= 1/?4?Dtexp(?L2/(4Dt))dt. (3.36)
By averaging RD(t) (equation [3.35]), over all possible times with the weighting
factor P(t) defined above10, it is possible to recover equation (3.34).
3.2.3 Contact resistance
Besides the spin-dependent resistance described in the previous sections, another
significant measurement is that in which the current is injected from lead B1 and
drained through lead N1, while the voltage between leads T1 and N2 is measured
(see Fig. 3.9). In the absence of spin injection the result of this measurement would
be the contact resistance Rc1, defined in equation (3.32). However, in the presence
of spin injection there is an additional effect produced by the spin accumulation at
the interface which acts as an additional resistance for the injection of spin polarized
electrons ?R [13, 53], and which was illustrated in the previous section in Fig. 3.8.
The general expression for the contact resistance is complicated and depends not
only on the parameters of the measured contact but also on the distance to the other
contact and its parameters. However, for large separations betweenF1 andF2, it can
be shown that ?R is always positive. This can be understood from the perspective
of the Johnson-Silsbee spin-charge coupling [27, 3, 24]. Spin that accumulates near
the FN interface due to the interplay of spin relaxation and spin diffusion acts as an
10More precisely, integraltext?
0 RD(t)P(t)dt.
49
effective impedance for the flow of more spin polarized electrons since it raises the
local electrochemical potential of the majority spin in both F and N metals. Since
spin and charge are carried by electrons, the back-flow of spin due to diffusion is
accompanied by an additional resistance to charge flow. An important limiting case
that yields a simple result is that of large contact resistances [equation (3.33)]. In
that case the measured contact resistance is given by equation 3.32.
3.3 Effect of leakage currents
The effect of leakage currents was analyzed by solving equations 3.27 in the general
case of additional current through regions 4, 5, and 7 in Fig. 3.9. The results of
the analysis are shown in Fig. 3.10 for five different values of the leakage current
through each of the regions. For the calculations a current of 100 ?A was applied
between leads T1 and N1 (see Fig. 3.9) and an injector-detector separation of 582
nm was used. For leakage currents that are more than 103 times smaller than the
injection current there is no significant change in the nonlocal resistance. When the
leakage current is as large as 1% of the injection current there is only a 6% change in
the nonlocal resistance, but above 1% the nonlocal resistance changes dramatically.
Therefore the effect of leakage currents is significant only for extremely large leakage
currents, larger than 0.1% of the injection current.
3.4 Hanle effect: spin precession
An important tool for extracting parameters such as the spin relaxation length and
the interfacial spin current polarization in spin valve structures is the Hanle effect,
50
1 10 100 1000 10000
ILeak (nA)
0
0.2
0.4
0.6
0.8
1
1.2
Nonlocal
Resistance
(m
?)
Figure 3.10: Nonlocal resistance for a current of 100 ?A in the presence of leakage
currents through regions 4, 5, and 7 in Fig. 3.9. The horizontal axis gives the
leakage current through each of the leads.
which is equivalent to the zero frequency version of Transmission Electron Spin
Resonance (TESR). In this section I will give a brief introduction to spin precession
in a constant magnetic field, and will explain how spin precession affects the nonlocal
resistance measurements in the case of ballistic and diffusive transport.
If an electron spin is originally in the state |?(t = 0)? = |+y?, that is, at time
zero the spin is ?up? in the y direction, then, in the presence of a magnetic field
of magnitude B? in the z direction, the electron spin at time t will be given by
|?(t)? = exp(?i?t?Sz/?h)|?(t = 0)? where ? is the Larmor frequency ? = g?BB?/?h,
?B is the Bohr magneton,gis the electron Lande g-factor, and ?Sz is the spin operator
in the z direction. After applying the exponential operator the result is the well
51
known expression [54]|?(t)? = cos(?t/2)|+y??sin(?t/2)|?y?. The probability of the
electron having spin up in the y direction after time tis therefore cos2(?t/2) and the
probability of the electron having spin down at timetis sin2(?t/2). The average spin
per electron in the y direction at time t is therefore ?h/2[cos2(?t/2)?sin2(?t/2)] =
?h/2cos(?t). So even though each electron spin precesses around the direction of
the magnetic field with frequency ?/2, the average spin in the y direction precesses
with frequency ?.
In the absence of spin relaxation and in zero magnetic field, spin polarized
electrons injected at F1 (see Fig. 3.9) diffuse to F2 without losing their spin infor-
mation11. Furthermore, if electron transport is ballistic betweenF1 andF2, all of the
electrons will take the same time to travel from the injector to the detector, say a
time tb. If the perpendicular magnetic field is zero then ? = ?tb = 0 and the average
spin in the y direction (see Fig. 3.9) does not change with time so the electron spins
arrive to the detector aligned with the magnetization of F2 and the measured non-
local resistance should be given by the positive value of equations (3.31) and (3.34).
When a nonzero perpendicular magnetic field is applied, the average spin in the y di-
rection of the electrons arriving to the detector is proportional to cos(?tb) [55, 49]12.
If the average spin in the y direction of the electrons that arrive atF2 has rotated by
an angle? = ?tb = ?, their spin is completely anti-aligned with the magnetization of
the detector, and the measured nonlocal resistance should be given by the negative
value of equations (3.31) and (3.34). If ? = ?/2 the average spin in the y direction
11For simplicity it can be assumed that both the magnetizations of F1 and F2 are aligned.
52
of the electrons arriving to F2 will be zero so the nonlocal resistance vanishes. If
the field is so strong or the time of flight so long that ? = ?tb = 2?, the spin of the
electrons arriving to F2 will once more be aligned with F2 giving again a positive
value. However, for values of the perpendicular magnetic field B? comparable to
the internal field of the magnetic material, the magnetization of F1 and F2 can start
to rotate out of the plane and further considerations have to be made to interpret
the results.
Transport between F1 and F2 is, however, diffusive, so there is a distribution
of times for the electrons to travel between injector and detector (equation [3.36]).
Furthermore, the spin relaxes with a characteristic length scale ?. Therefore, in
the limit of large contact resistances, the nonlocal resistance should be given by
a weighted sum of equation 3.35 times the rotation term cos(?tb) with the weight
being given by the probability density P(t), more precisely
R(?) = ?
integraldisplay ?
0
RD(t)cos(?t)P(t)dt. (3.37)
The complete expression in the case of large contact resistances is
R(?) = ?
integraldisplay ?
0
1
2
P1P2RN?
4?Dt exp(?Dt/?
2)cos(?t+?)exp(?L2/(4Dt))dt, (3.38)
where cos(?t) has been replaced by cos(?t+alpha) to allow for possible misalign-
12This factor of cos(?tb) can also be understood as a position dependent rotation of the reference
frame, so that at each coordinate x the positive y axis is aligned with the average spin of the
electrons (the reference frame rotates with frequency ? around the axis of the magnetic field).
Therefore F2 will be rotated by an angle ?tb with respect to the local reference frame, and the
projection of the average y spin onto the magnetization direction of F2 will be given by cos(?tb).
53
ments between the magnetizations of F1 and F2 in case that an angle ? exists
between the easy axis of the two magnetic contacts.
3.5 Interface spin-flip scattering
Throughout the previous sections of this chapter it was assumed that spin-flip scat-
tering at the F-N interfaces was negligible. In this section the effect of interfacial
spin-flip scattering will be considered by relaxing the assumption of spin current
conservation at the interfaces. To quantify the amount of interfacial spin-flip scat-
tering, parameters which can in principle be calculated if the details of the spin
scattering are known, will be used. Spin valves with both transparent and resistive
interfaces can be analyzed using the general framework of resistive contacts, since
the former can be considered to be a limiting case in which the interface resistances
vanish.13
3.5.1 Nonlocal geometry spin valve with interfacial spin-flip scattering
The process of interfacial spin-flip scattering can be pictured as the transition from
the incoming states |?? > and |?? > in the nonmagnetic metal into the outgoing
states |?? > and |?? > in the ferromagnet, where ? and ? refer to the spin in the z
direction. The rate at which transitions between incoming and outgoing states occur
can be calculated using Fermi?s golden rule if the density of states and the transition
probabilities between states |?? >, |?? > and |?? >, |?? > are known. This requires
knowledge of microscopic details of the interfaces which are outside the scope of
13See discussion after equation (3.31).
54
,M
,m
,M
,m
,M
,m
,M
,m
,m
,M ,m
,M
Mm*1/R mM*1/R mm
*1/R
MM*1/R
1/R mm
1/R MM
1/R MM
1/R mm
Mm*1/R mM*1/R
1/R mM
1/R Mm
1/R mM
mm*1/R MM*1/R
1/R Mm
(d) FN
(a) FN
(c) FN
(b)
,M
,M
,m
FN
,m
Figure 3.11: Diagrams representing the scattering at the interface together with
phenomenological coefficients R that represent the interface resistance of each spin
channel. (a) Majority carriers of each region are aligned and have spin ?. (b) Major-
ity carriers of the ferromagnet have spin?while majority carriers of the nonmagnetic
metal have spin ?. (c) Majority carriers of each region are aligned and have spin ?.
(d) Majority carriers of the ferromagnet have spin ? while majority carriers of the
nonmagnetic metal have spin ?.
this work. However, in the case of small voltage biases14 [56, 57] the currents and
electrochemical potentials at both sides of the interface can be related by means
of a set of phenomenological parameters with the physical meaning of resistance
which completely describe the scattering process of the transmitted electrons. The
pictorial representation shown in Fig. 3.11 is useful for defining each of the resistance
parameters R. The horizontal lines represent conserved spin transitions while the
lines drawn at an angle represent spin-flip events at the interface. ? and ? clearly
label spin up and spin down channels, which have an additional label M for spin
majority or m for spin minority. This labels can be used for both the ferromagnet
14Small compared with the thermal energy divided by the charge.
55
and nonmagnetic metal.15 N and F label the nonmagnetic and ferromagnetic sides
of the interface. Figures 3.11 (a), (b), (c), and (d) have different alignments between
the excess magnetization in the nonmagnetic metal and the magnetization in the
ferromagnet. In Fig. 3.11 (a) the majority carriers of each region have spin up,
while in Fig. 3.11 (b) the majority carriers in the nonmagnetic metal have spin
down. In Fig. 3.11 (c) the majority carriers of each region have spin down, while
in Fig. 3.11 (d) the majority carriers in the nonmagnetic metal have spin up. A
resistance term such asR?Mm is defined as the interfacial resistance from the majority
(M) spin channel in the nonmagnetic metal to the minority (m) spin channel in the
magnetic metal with spin-flip (*). On the other hand a term such as RMm means
the interfacial resistance from the majority (M) spin channel in the nonmagnetic
metal to the minority (m) spin channel in the magnetic metal without any spin-flip.
Therefore subindexes like Mm do not necessarily require a spin flip since majority
carriers in one metal can have spin ? while majority carriers in the other metal can
have spin ? [Fig. 3.11 (b)].
From the diagrams shown in Fig. 3.11 it is possible to write the boundary
conditions for the electrochemical potential at each F-N interface, which relate the
difference in electrochemical potential across the interface with the current and
with the resistance coefficients R. As an example of this procedure, the boundary
equations for the F2-N interface (detector) when the magnetizations of injector and
15Majority and minority carriers are well defined in ferromagnets. We have extended these terms
to refer to the nonequilibrium magnetization present in the nonmagnetic metal, so if there is an
excess of electrons with spin up, then those are the majority carriers in the normal metal.
56
detector are aligned and pointing ? [Fig. 3.11(a)] are written explicitly:
jF? AC2 = (?N? ??F? )/RMM + (?N? ??F? )/R?mM
jF? AC2 = (?N? ??F? )/R?Mm + (?N? ??F? )/Rmm
jN? AC2 = (?N? ??F? )/RMM + (?N? ??F? )/R?Mm
jN? AC2 = (?N? ??F? )/R?mM + (?N? ??F? )/Rmm, (3.39)
where AC2 is the contact area of the F2-N interface, and wherejF? (jN? ) is the current
density with spin ? going into the ferromagnetic region (out of the nonmagnetic
region). The boundary conditions at the F2-N interface for other configurations of
the magnetizations of the injector and detector can be deduced from Fig. 3.11(b),
(c), and (d). For the F1-N (injector) interface, where current is injected from the
ferromagnetic to the nonmagnetic metal, or in general when current flows in the
opposite direction, diagrams similar to those shown in Figs. 3.11(a) and (c) can be
obtained by simply reversing the direction of the arrows. For the injector interface
only such diagrams exist since the majority carriers in both F1 and N always have
the same spin orientation. If the magnetizations of the injector and detector are
parallel, then diagrams (a) or (c) should be used for both injector and detector
interfaces. However, if the magnetization of injector and detector are anti-aligned
either diagrams (a) for the injector interface and (d) for the detector interface or
diagrams (c) for the injector interface and (b) for the detector interface must be used.
Diagrams (b) and (d) therefore only apply to the F2-N (detector) interface, where
57
majority carriers at both sides of the interface can have different spin orientation
when the magnetization of the injector is antiparallel to the magnetization of the
detector.
In order to see the effects of spin scattering, and without loss of generality,
it will be assumed that the magnetization of the injector is fixed and pointing ?,
while the magnetization of the detector can have two possible orientations, either ?
or ?. Therefore, according to the previous discussion, when both magnetizations are
aligned and pointing ?, diagram (a) should be used to find the boundary conditions
for both interfaces. On the other hand, when the magnetization of the injector is
reversed, diagram (a) should be used for the injector interface and diagram (d) for
the detector interface. After writing the boundary conditions corresponding to these
diagrams, and using them to solve equations (3.25), an expression for the nonlocal
resistance can be found. Since the result is not very illuminating it will not be
transcribed here. However, in the limit of contact resistances much larger than the
resistance of the ferromagnetic and nonmagnetic metals over their spin relaxation
length [see equation (3.33)], it is possible to simplify the expression for the nonlocal
resistance to obtain16
R?? = 12P1P??2 RN exp(?L/?)
R?? = 12P1P??2 RN exp(?L/?), (3.40)
where P of the injector and detector in the presence of interface spin-flip scattering
16See equation 3.34.
58
are defined as
P1 = (1/R1,MM ?1/R1,mm) + (1/R
?
1,Mm ?1/R
?
1,mM)
(1/R1,MM + 1/R1,mm) + (1/R?1,Mm + 1/R?1,mM)
P??2 = (1/R2,MM ?1/R2,mm) + (1/R
?
2,Mm ?1/R
?
2,mM)
(1/R2,MM + 1/R2,mm) + (1/R?2,Mm + 1/R?2,mM)
P??2 = ?(1/R2,mM ?1/R2,Mm) + (1/R
?
2,MM ?1/R
?
2,mm)
(1/R2,mM + 1/R2,Mm) + (1/R?2,MM + 1/R?2,mm) . (3.41)
The difference between R?? and R?? is due to the difference between P??2 and P??2 ,
which in turn comes from the difference between the spin-flip scattering of major-
ity and minority carriers in the nonmagnetic metal. For both alignments of the
magnetizations of F1 and F2, the spin polarization of the injector (P1) is the same.
In the absence of interfacial spin-flip scattering all the resistance terms in
equation (3.41) that contain a ?*? should be dropped and the result should be the
same as the original definition of the Pi given by equation (3.23). This equivalence
can only be obtained if Ri? = Ri,MM = Ri,mM and Ri? = Ri,mm = Ri,Mm, that is, if
the non-spin-flip resistances are equivalent for majority and minority carriers in the
nonmagnetic metal. Therefore equation (3.23) is a particular case of equation (3.41)
with the assumption that the tunnelling of spin up and down electrons depends on
which channel is the majority carrier in the ferromagnet, but is independent of
which channel is the majority carrier in the nonmagnetic metal. This assumption
can be justified since the difference in wave functions, the density of states at the
Fermi energy, and the Fermi momenta between spin sub-bands is much larger in the
ferromagnet. Hence the small differences between majority and minority carriers in
59
the nonmagnetic metal can be ignored for non-spin-flip tunnelling processes. If this
assumption also holds for the spin-flip resistance terms, the nonlocal resistance for
anti-aligned magnetizations of F1 and F2 is the negative of the nonlocal resistance in
the case of aligned magnetizations. If this assumption does not hold, an additional
component of the nonlocal resistance, which does not change sign when the injector
magnetization is reversed, appears. An example of a case in which the assumption is
not valid is when the probability of a spin-flip event for spin ? is proportional to the
density of states with spin?inthe ferromagnet. According to Fermi?s golden rule the
transmission rate is proportional to the density of states of the outgoing spin state ??
(opposite to?) and to the probability of the transition. Therefore the product of the
densities of states for majority and minority carriers in the ferromagnet appear in the
calculation of all of the spin-flip resistances R?. In this case the differences between
minority and majority carriers in the nonmagnetic metal might be important so the
approximation is not necessarily valid. This will be relevant in chapter 5, where
experimental evidence for the existence of an additional component of the nonlocal
resistance, which does not change sign with the reversal of the magnetization of the
injector, is shown.
60
Chapter 4
Sample fabrication and experimental setup
In this chapter the techniques used to fabricate the spin valve devices used for spin
injection and detection measurements via transport will be described. Then a brief
overview of the experimental setup will be given.
4.1 Sample fabrication
Since the size of the smaller features required for the spin valve devices is of the
order of 100 nm, and since the alignment between magnetic and nonmagnetic levels
is essential, the samples are fabricated using two levels of electron beam lithography
together with thermal evaporation. The complete procedure will be explained in
this section.
One inch 100-orientated Si wafers with 500 nm thermally grown SiO2 are used
as a substrate. The wafers are covered with a bilayer of positive e-beam resist [poly
methyl methacrylate (PMMA)] of molecular weights 150,000 and 360,000 by spin-
ning at 2500 rpm, and have typical thickness 150 nm and 250 nm respectively. Each
layer of PMMA is baked at 40oC for 2 minutes, 80oC for two more minutes, and
140oC for 30 minutes to avoid inter-layer mixing. A final bake of 90 minutes is done
after the bilayer is completed. The coated wafers are cleaved into quarters, each of
61
which is processed separately. About ten devices can be fabricated in each of the
wafer quarters. An electron beam with an accelerating voltage of 30 kV, working
distance of 8mm, typical currents 30-60 pA, and field scale 60?m1, is used to expose
the regions of the PMMA that are to be covered by the magnetic material with a typ-
ical dose of 5.9 pC/?m2 [Fig. 4.1(a)]. The difference in molecular weight, together
with the backscattered and secondary electrons creates a profile in the PMMA as
shown in Fig. 4.1(b). In order to align the first and second levels of electron beam
lithography, and in order to focus the electron beam correctly, a small region of
PMMA is overexposed at a field scale of 8?m with the typical e-beam currents for
30-45 seconds. The region with cross-linked PMMA (due to overexposure) can be
easily seen with the SEM, so focusing and stigmation adjustments can be made. If
immediately after one of these focusing spots has been made the areas to be covered
by the magnetic material are exposed by the electron beam, this focusing spot can
be used as a very precise alignment marker for the second level of lithography [see
Fig. 4.2(a)]. Developing, or removal of the exposed PMMA, is done by immersing
the wafer in a methyl isobutyl ketone (MIBK):isopropyl alcohol (IPA) (1:3 by vol-
ume) solution at 23o C for 30 seconds [Fig. 4.1(c)], rinsing the wafer by immersing it
in isopropanol, and blow drying it with research grade compressed N2. This creates
a mask for transferring the patterns onto the substrate. The wafer is immediately
transferred to the vacuum chamber of an evaporator to avoid contamination. After
reaching typical base pressures between 7?10?7 Torr and 2?10?6 Torr, the wafer
1JEOL-420 Scanning Electron Microscope (SEM)
62
000000000000000
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000000000000000
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0000
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00000
00000
00000
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11111
11111
11111
11111
(c)
(a)
(f)
(e)(b)
(d)
150K PMMA
360K PMMA
Substrate
e?beam
e?beam
MIBK:IPA
ion?beam
metal
Warm acetone
Figure 4.1: Schematic of the e-beam lithography and thermal evaporation process
for sample fabrication. A side view of the wafer is shown. (a) Substrate and PMMA
bilayer. (b) Exposure of the PMMA to the e-beam. (c) Developing (or removal)
of the exposed PMMA. (d) Ion milling of the substrate for cleaning and increasing
adhesion. (e) Thermal evaporation of metal. (f) Lift-off of the remaining PMMA
and metallization.
63
is ion milled with Argon for 30-80 seconds in a Commonwealth Plasma system at a
pressure of 1?10?4 Torr, with a beam voltage of 500 V, an accelerating voltage of
100 V, 5 mA of current, and a neutralizer current of 7.5 A [Fig. 4.1(d)]. The purpose
of ion milling is to clean the substrate of any remaining undeveloped PMMA and
additional contamination (like IPA), increasing the adherence of the metallic film.
The vacuum chamber is pumped once more to a pressure lower than 2?10?6 Torr,
and the magnetic material [Co or Py (permalloy2)] is thermally evaporated using
rates between 0.1 and 0.5 nm/s. The typical thickness of the evaporated film is 36
nm according to the in-situ film thickness monitor, which has been calibrated using
a Tencor alfa-step profilometer. The evaporated material sticks to the exposed wafer
through the holes in the mask and also to the PMMA mask itself [Fig. 4.1(e)]. After
waiting for the system to cool down for 30 minutes at low pressure, the sample is
extracted from the chamber and immediately introduced in warm acetone, which
lifts-off the remaining PMMA removing the film from unexposed regions and leaving
only the metal that adhered directly to the substrate. Ultrasound is sometimes used
to help with the lift-off process [Fig. 4.1(e)].
After the first level has been completed, the sample is again covered with a
bilayer of PMMA, baked and inserted in the SEM. Some of the focusing spots made
during the first level are used to align the rotation of the stage, and also as a map to
find thepositionof each sample. The electron beamis blocked andthestage ismoved
to the focusing spot that serves as the alignment marker for a particular sample.
2NiFe 80-20.
64
The beam is unblocked at a small field scale (10?m) and after coarse adjustment
has been made, the field scale is reduced to 1?m, where precise alignment to within
50 nm to 100 nm can be made. Then, electron beam lithography is done at two
different magnifications. First the small features are exposed to the e-beam with
the same parameters as in the first level [see the narrow wires in Fig. 4.2(a)], but
then a field scale of 1.5 mm is used to expose the large features such as connecting
wires and bonding pads using currents as large as 7000 pA [see Fig. 4.2(b)]. In
the case of transparent samples, the remaining steps of the second level are exactly
the same as the first, except that Cu is evaporated instead of Co or Py using a
different mask. In the case of resistive contacts, additional steps in the second level
of material deposition, between the ion milling [Fig. 4.1(d)] and the Cu thermal
evaporation [Fig. 4.1(e)], are necessary. First, just after ion milling, when the
pressure is lower than 2?10?6, 2 nm of Al are thermally evaporated to make a thin
covering of the exposed magnetic material. Pumping is then stopped and 100 to 800
mTorr of O2 is let into the chamber. The aluminum is oxidized for 4 minutes, after
which pumping is resumed, and the remaining steps of the second level [Figs. 4.1
(e) and (f)] are completed. By controlling the oxygen pressure, the barrier thickness
and resistivity can be controlled. However, the resistance of the barrier is very
sensitive to its thickness, which can vary due to surface roughness of the magnetic
film. Therefore fabricating barriers of specified resistance can be challenging, but
since the only requirement is that they be large compared to the resistance of Py,
Co, and Cu over their spin relaxation lengths (see chapter 3), any value larger than a
65
(a)
(b)
Figure 4.2: SEM micrographs of the typical large field samples design. (a) Focusing
and alignment spot, together with additional wiring at a field scale of 120 ?m. (b)
Bonding pads and additional wiring at a field scale of 2mm.
66
few ohms is acceptable. However, due to the stray capacitances in the experimental
setup, barrier resistances larger than a few hundred ohms contribute to capacitive
leakage which increases as the square of the barrier resistance and should therefore
be avoided.
Spin valve designs evolved as the understanding of the spin resistance in-
creased. The first working sample design is shown in the SEM micrograph in Fig.
4.3(a). This sample is of the cross geometry type. The light gray lines are made
of Cu, the gray ones are made of Py, and the remaining dark gray regions are the
substrate. Vertical alignment is critical since the separation between the two Py
lines is close to 350 nm and the width of the horizontal Cu line is 100 nm, leaving
only 125 nm of clearance above and below, much smaller than the typical backslash
of the SEM mechanical stage. However, as explained in chapter 3, this particular
geometry is not optimal for spin injection and detection experiments, and a nonlocal
geometry [Fig. 4.3(b)] is preferred. The micrograph shown in Fig. 4.3(b) was taken
at an angle to show the crossings of the Cu lines over the magnetic contacts. A
better view of a single contact is shown in Fig. 4.3(c) where a very nice overlap
between the magnetic and non-magnetic level can be seen. For some of the mea-
surements samples with many consecutive magnetic contacts that would be used
for both injection and detection were fabricated [Fig. 4.3(d)]. In such samples the
distance between consecutive magnetic contacts increases so the length dependence
of the nonlocal resistance can be studied with sets of measurements on a single
sample. Figures 4.3 (a), (b), and (d) show two essential features of the magnetic
67
(a)
(b) (d)
(c)
(Co)
Cu
Cu
Cu
(Co)
?
(Py)
(Py)
2
Cu
(Co)
(Py)
1F
Al  O3
1   m
1F
2F
2F
1
F
F
Figure 4.3: SEM micrographs of finished spin-valve samples. (a) Cross geometry
Py-Cu sample.(b) Typical nonlocal resistance Co-Cu sample. (c) Typical contact
between magnetic and non-magnetic levels for a Py-Cu sample. (d) Sample with
many magnetic contacts for measuring the length dependence of the nonlocal resis-
tance.
contacts. First, the high aspect ratios that were used to keep the magnetization
of the magnetic contacts in either of two antiparallel directions along the longer
(easy) axis of each contact. Second, the different aspect ratios that were used for
consecutive magnetic injector-detector pairs in order to allow independent control
of the magnetization alignment of each contact by means of an external magnetic
field.
For completeness, tables with the relevant information about the samples that
were reported in this thesis are shown below.
68
Device Name Configuration Py Co Al2O3
sam2b8s1 Cross Geometry,2 ? None
sam4b1s4 Nonlocal,8 ? None
sam10b3p1 Nonlocal,9 ? None
sam10b3p3 Nonlocal,9 ? None
sam12b1p2 Nonlocal,9 ? None
s sam11b2y1p2 Nonlocal,2,2 ? 410mTorr
s sam11b2y3p2 Nonlocal,2,2 ? 122mTorr
s sam12p2 Nonlocal,2 ? 423mTorr
s sam12p11 Nonlocal,2 ? None
s sam10b1p1 Nonlocal,9 ? None
s sam10b3y3p1 Nonlocal,9 ? None
Table 4.1: Device characteristics. The number of magnetic contacts is given under
?Configuration?, where ?2,2? means two contacts separated by a large distance from
the next two contacts. The magnetic material used for each device is specified with
a check under either ?Py? (Permalloy) or ?Co? (Cobalt). If an oxide barrier is used
between ferromagnetic and nonmagnetic metals, the pressure ofO2 during oxidation
is given under ?Al2O3?.
When the samples have been completed, they are again coated with PMMA
for protection, but this time no baking is used since this affects the quality of the Cu
film. A manual wafer dicer is then used to separate the samples that fit into a wafer
quarter, after which they are dipped in acetone to remove the PMMA protective
layer, rinsed with isopropanol, and blow dried. Silver paint is used to adhere each
finished sample to a ceramic header with 16 Be-Cu pins, and after grounding the
pins to avoid electrostatic discharges that can damage the thin lines in the device,
bonds with 32?m thick aluminum wire are made from the pins to the bonding pads.
4.2 Experimental setup
The experimental setup has four main components which are the probe, the control
electronics, the measurement electronics, and an AD-DA converter and computer.
69
Device Name Th W Cu Th Cu W L
sam2b8s1 85 nm 470nm, 120nm3 90 nm 120nm2 35 nm2
sam4b1s4 30 nm 470nm,100nm 48 nm 150nm 200-1500nm3
sam10b3p1 34 nm 135 nm 45 nm 125 nm 354-1068nm4
sam10b3p3 34 nm 140 nm 45 nm 125 nm 354-1068nm4
sam12b1p2 48 nm 100 nm 76 nm 130 nm 354-1068nm4
s sam11b2y1p2 32 nm 100 nm 54 nm 100 nm 525,740 nm
s sam11b2y3p2 32 nm 115 nm 54 nm 120 nm 987,1220 nm
s sam12p2 36 nm 150 nm 54 nm 150 nm 717 nm
s sam12p11 36 nm 128 nm 54 nm 180 nm 980 nm
s sam10b1p1 36 nm 130 nm 48 nm 140 nm 252-966nm5
s sam10b3y3p1 42 nm 130 nm 48 nm 140 nm 252-966nm5
Table 4.2: Device lengths. The abbreviated column titles are ?Thickness? of the fer-
romagnetic material, ?Width? of the ferromagnetic material, ?Cu Thickness?, ?Cu
Width?, and ?L?, the separation between contacts. In the case of many magnetic
contacts the separations of consecutive contacts are given in the footnotes.
The probe consists of an aluminum shielding box with BNC connectors, fixed on
top of a copper-nickel tube with a brass ring at the bottom. A nonmagnetic 16 pin
socket which holds the ceramic sample header is thermally grounded and fixed to
the brass ring with Emerson-Cummings 2850 epoxy. A Lake-Shore platinum (PT-
111) calibrated thermometer is also thermally grounded and fixed to the brass ring.
Wiring for the socket is made of copper-nickel wire with a resistance of 23 ? from
the box to the socket, while low resistance copper wiring is used for the magnet
and thermometer leads. Wires are paired up, twisted and covered with additional
insulation. In addition, signal wires are tied together to reduce inductive pickup.
However this increases their capacitance, which is 230 pF between twisted pairs,
120 pF between non-twisted pairs, and 160 pF between each lead and ground. The
2Quantities not measured but expected from lithography or from similar measured samples.
3Separations between consecutive lines are 200, 240, 240, 402, 798, 1200, 1500 nm.
4Separations between consecutive lines are 354, 456, 558, 660, 762, 864, 966, 1068 nm.
5Separations between consecutive lines are 252, 354, 456, 558, 660, 762, 864, 966 nm.
70
superconducting magnet (made of niobium in a copper matrix) is a small 1.25 inch
outer diameter by 2 inch high cylinder with a soft iron core in part of its interior,
which provides 57.3 mT/A at the site of the sample in a direction parallel to the
axis of the probe. For measurements in which a field parallel to the plane of the
sample is needed, an additional socket is used, which allows the plane of the sample
to be set vertically. In such a configuration the field produced by the magnet is 47.1
mT/A at the site of the sample.
The control electronics is used to measure the temperature close to the sample
and to set and sweep the magnetic field. Measurement of the temperature is done by
using a Monogram Omega controller, which measures the four terminal resistance of
the Pt thermometer, converts it to a temperature value using a previous three point
calibration and interpolation in the 77 K-293 K range, and outputs a voltage propor-
tional to the temperature which is read by the computer via the AD converter. The
magnetic field control is done by using an HP 3325A synthezizer/function generator
to produce a triangular wave of adjustable frequency (typically in the 0.01-0.00003
Hz range) and amplitude which is used to voltage control a bipolar Kepco source
(model BOP 20-10 M) operating as a voltage source. The output of the source is
connected to a 0.101 ?, 25 W resistor, at which the voltage drop is measured using
an HP3456A digital multimeter and the AD converter in order to monitor the cur-
rent. After the resistor, a coaxial cable is used to carry the current to an isolated
BNC on the box of the probe, and from there thick Cu wires carry the current down
to the magnet.
71
The measurement electronics is used to bias the spin-valve device and detect
the small signals that are produced. To accomplish this, two PAR-124 lock-in am-
plifiers (I and II), tuned to a frequency of 11.1 Hz are used. Operating at higher
frequencies was avoided even though it could reduce the noise even further, since
it would increase the capacitive leakage through the wiring and through the input
impedance of the lock-in amplifiers. The second lock-in amplifier is phase locked
to the internal oscillator of the first one. The source output of the first lock-in is
connected to a Vishay bias resistor of 100K? in series with a Vishay 2K? resistor to
monitor the current. The bias signal is then taken by a coaxial cable to one of the
BNC connectors in the shielding box at the top of the probe (connected, say to lead
B1 in Fig. 3.9). The voltage drop at the 2K? resistor is measured with that same
lock-in (I) and the output of I is read by the computer through the AD converter.
A grounding cap is used to connect the inner conductor (connected, say to lead N1
in Fig. 3.9) of another of the BNC?s to the shielding box (which is grounded only
through the exterior conductor of the current input coaxial cable). Two other coax-
ial cables are used to connect the remaining measurement leads (T2 and N2 in Fig.
3.9) to the inputs of the differential preamplifier of lock-in (II), and the in-phase
signal is monitored by the computer via the AD converter.
Experiments at low temperature are done by inserting the dipping probe in
liquid helium (for measurements at 4.2o K) and liquid nitrogen (for measurements at
77o K). Temperature variation in the 100o K-293o K range is achieved by inserting
the dipping probe in a dewar above the liquid nitrogen level. By changing the height
72
of the sample above the liquid nitrogen, or equivalently by allowing the nitrogen
level to slowly drop due to N2 evaporation, the temperature of the sample can be
adjusted. The dipping probe has a large thermal mass surrounding the sample so
the temperature changes in the device are slow to make sure that the temperature of
the thermometer and the temperature of the sample are equivalent. By measuring
the temperature dependence of any quantity both on cool-down and warm-up and
comparing theresults, variationsbetween thesample andthermometer temperatures
can be detected. During the experiments the temperature change was slow enough
so that the traces on cool-down and warm-up were equal to within the noise.
73
Chapter 5
Spin Valve Measurements
In this chapter I will present the measurements that were performed on Py-Cu and
Co-Cu spin valves, which include the characteristic switching of the spin-dependent
resistance with the reversal of injector or detector magnetization, spin precession
(Hanle effect), length dependence of the spin-dependent resistance, and temperature
dependence of the nonlocal resistance for aligned and anti-aligned configurations of
the injector and detector magnetizations. I present evidence for the existence of a
symmetric component of the nonlocal resistance and provide cross checks that were
done to rule out spurious origins of this previously ignored signal, like capacitive
leakage, heating coupled with thermo-power, and classical electrostatic effects com-
ing from the solution of Laplace?s equation for the potential in a non-ideal conductor
for a particular geometry.
5.1 Cross geometry configuration
The initial configuration that was used to study spin injection and detection was the
cross geometry described in chapter 3, Fig. 3.3 (see also the insets of the following
figures). In this section I will present the results of such measurements and will
describe the possible problems that may appear when interpreting the results.
74
? 100 ? 50 0 50 100
B|| (mT)
24
24.1
24.2
24.3
24.4
24.5
24.6
Spin
?dependent
resistance
(m
?)
F1 F2
I
V
000
000
000
000
111
111
111
111
000
000
000
000
111
111
111
111
B1 BN
T2
L 2N
T
L 1
0
0
1
1
0
0
1
1
Figure 5.1: R vs Bbardbl for sample sam2b8s1 at 4.2 K. This sample has transparent Py-
Cu contacts. The inset shows a schematic of the sample layout and measurement
configuration.
As discussed in chapter 3, one of the characteristic signals of spin injection is
the switching of the spin-dependent resistance defined asR = VTN?T2/IB1?BN (where
the labels are shown in the inset of Fig. 5.1) when the magnetization of either of the
magnetic contacts changes direction due to the applied field. Figure 5.1 shows the
spin-dependent resistance as a function of the magnetic field Bbardbl applied parallel to
the easy axis of the magnetic contacts of a Py-Cu sample with transparent interfaces
at 4.2 K. The solid (dashed) line is for increasing (decreasing) Bbardbl. At large negative
fields the injector (F1) and detector (F2) are parallel and pointing down. When the
field is increased above the positive coercive field of the injector, about 15 mT, its
magnetization reverses and R switches from 24.02 m? to 24.62 m?. When the field
75
is further increased above the positive coercive field of the detector, about 47 mT,
its magnetization reverses, the two magnetic contacts become parallel once more
but this time pointing up, and R switches to 24.04 m?, close to the large negative
field value. If the field is then decreased, the two magnetic contacts will remain
parallel until the negative coercive field of the injector is reached at -10 mT, when
its magnetization reverses and R switches to 24.60 m?. When the field is further
reduced to the negative coercivity of the detector, about -47 mT, its magnetization
also reverses, the two magnetic contacts point down, and R changes to its low value
of 24.02 m?.
The positive and negative field coercivities of the injector are different showing
that there is a preferred direction for its magnetization. In addition, the positive
field reversal of the injector is slower than the negative field reversal, providing more
evidence that it takes more energy to set the injector pointing up then down. This
can be expected since for this sample the injector is 2.2 ?m long by 0.5 ?m wide,
which gives a not very large aspect-ratio of 4.4, so any structural anisotropy arising
from the thermal evaporation is relevant1. Another important detail from Fig. 5.1
is the fact that the values of R for the injector pointing up and down are slightly
different (0.02 m?). This slight difference in R can be explained, not by a difference
in the spin current polarization of the injector (P1) for up and down directions
of its magnetization, but instead by a slight misalignment of the magnetization
1No field was applied during evaporation so the easy axis of different grains are not necessarily
aligned. For a small number of grains the distribution is not random and therefore a preferential
direction for magnetization might occur.
76
of F1 and F2, by anisotropic magnetoresistance (AMR), or Hall effect. The fact
that the value of R can be affected by AMR and Hall effect shows that this is
not the optimal geometry for the detection of a spin-related signal. In addition,
the existence of a large background (24.3 m?), roughly 40 times bigger than the
spin-related signal, creates drifts due to temperature changes in the electronics, and
denies the possibility of testing the symmetry of the spin-dependent signal that is
predicted by the model [see equation (3.20)].
For completeness, 77 K measurements of the same sample are shown in Fig.
5.2. Many traces are plotted simultaneously to show the reproducibility in the mag-
netization reversal and in the switching of R. The same features as those shown
for the 4.2 K measurement are still present, but the magnitude of the signal has
changed. The size of the steps decreased to about 0.48 m? (from 0.58 m? at 4.2
K), and the background resistance increased to 37.44 m?. Both trends are expected
since as temperature increases both spin preserving and spin-flipping mechanisms
become more effective, decreasing the spin diffusion length (and therefore the mag-
nitude of the steps) and decreasing the mean free path, (enhancing the resistivity
which gives the main contribution to the background resistance).
In conclusion, spin injection and detection was shown to be possible in meso-
scopic Py-Cu spin valves, the reversal of the magnetic contacts and the switching
of the spin-related signal were shown to be reproducible, and the correct tempera-
ture trend for the background resistance and for the spin-dependent resistance was
observed. However, since AMR, Hall effect, and even ohmic voltages are part of the
77
? 60 ? 40 ? 20 0 20 40 60 80
B|| (mT)
37.2
37.3
37.4
37.5
37.6
37.7
Spin
?dependent
resistance
(m
?)
F1 F2
I
V
000
000
000
000
111
111
111
111
000
000
000
000
111
111
111
111
B1 BN
T2
L 2N
T
L 1
0
0
1
1
0
0
1
1
Figure 5.2: R vs Bbardbl for sample sam2b8s1 at 77 K. This sample has transparent Py-
Cu contacts. The inset shows a schematic of the sample layout and measurement
configuration.
measured signal, this geometry is problematic and should not be used to extract
parameters such as the spin diffusion length and the degree of spin polarization.
5.2 Nonlocal geometry configuration
As shown in the previous section the cross geometry is difficult to interpret theo-
retically since it includes different physical effects. To solve this problem a different
geometry was used for the rest of the experiments. The new geometry has the very
appealing feature that it is nonlocal, so none of the AMR, Hall effect, or ohmic
voltages are part of the measured signal. That is, in the absence of spin injection
the expected signal should be exactly zero2. I will begin this section by present-
2This is not entirely correct, as will be explained later in this chapter when talking about
possible artifacts. Local heating coupled with thermoelectric effects can generate spurious voltages,
78
ing results of the nonlocal resistance as a function of separation between magnetic
contacts, from which the spin diffusion length of electrons and the degree of spin
polarization can in principle be extracted. Then I will show how particularities of
each of the ferromagnetic-nonmagnetic metal contacts can affect the extraction of
such quantities, and how a slightly different experiment, based on spin precession
(Hanle effect) can be used to extract these numbers with a single measurement. I
will then give evidence for the existence of increasing interface spin-flip scattering
with temperature, which has been previously unaccounted for, and which appears as
a temperature dependent symmetric component of the nonlocal resistance. Finally
I will provide additional measurements that were performed to rule out artifacts.
5.2.1 Length dependence of the nonlocal resistance
The nonlocal measurement configuration is depicted in the inset of Fig. 5.3 and
was discussed in detail in chapter 3. The nonlocal resistance, defined as RNL =
VT2?N2/IT1?N1, can have either of two values, depending on the alignment of the
injector and detector magnetizations, R??NL for aligned and R??NL for anti-aligned
magnetic contacts [see Eq. (3.29)]. The difference in the value of the nonlocal
resistance between the two possible alignments divided by two,
RA =
bracketleftBig
R??NL ?R??NL
bracketrightBig
/2, (5.1)
the finite input impedance of the amplifiers (lock-in) as well as capacitive leakage between leads
can generate small sample dependent background voltages, and wide overlap areas between the
magnetic contacts and the normal metal wire can generate small decaying potential drops that
can in principle affect the signal. It will be shown, however, that this was not the case in our
measurements.
79
0 0.2 0.4 0.6 0.8 1 1.2
L (?m)
0
0.05
0.1
0.15
0.2
0.25
R A
(m
?)
F2
F1
N 21N
T1 T2
I V
0000
0000
0000
0000
0000
1111
1111
1111
1111
1111
00
00
00
00
00
00
11
11
11
11
11
11L
0
0
1
1
0
0
1
1
=900 nm?
Figure 5.3: Nonlocal resistance switching (RA =
bracketleftBig
R??NL ?R??NL
bracketrightBig
/2) as a function of
the separation L between injector and detector for sample sam4b1s4, with transpar-
ent Py-Cu contacts at 4.2 K.
is related to the spin diffusion length ? by means of equation (3.29), as discussed in
the previous chapter. In order to extract the spin diffusion length of electrons in a
nonmagnetic metal, samples withvarying separationLbetween injector anddetector
are fabricated, and the value of RA is recorded as a function of the separation L.
The typical sample layout for these measurements is shown in Fig. 4.3(d), where
consecutive magnetic contacts were used as injector-detector pairs.
Figure 5.3 shows the length dependence of RA for sample sam4b1s4, together
with a fit to equation (3.29). This sample has 25 nm thick Py and 48 nm thick
Cu lines, and 60 seconds of ion milling with 10?4 Torr Ar at 250 V and 5 mA
current. The data shows the correct trend of decreasing RA with increasing L.
80
The extracted value of the spin diffusion length is 900 nm, which is comparable
to previous measurements by different groups [22, 58]. However, a better compar-
ison can be established by calculating the ratio between the momentum and spin
scattering times (ap = ?p/tau). For our data ap =2.7?10?4, while for other groups
ap = 6.9?10?4 [22] and ap =3.2?10?4 [58]. It must be pointed out that in order to
extract ?, the value of the spin diffusion length of electrons in the magnetic material
must be known, and since it cannot be measured for the spin valve wires, a reason-
able value measured for a different system has to be used [59, 60, 61]. To obtain
the best fits shown in Figs. 5.3 and 5.4 we used values of the spin diffusion length
in Py between 3nm and 5nm. This is not the only problem. The data is scattered
so it cannot be fit well by the model, and since the fitting equation is nonlinear,
there is a large set of parameters that can fit the data and have a similar mean
square deviation. What this means is that the confidence intervals are large, so the
certainty with which it can be claimed that the extracted value of ? is correct is
very low.
Figure 5.4 shows two more sets of measurements for samples with different
thickness of the Py and Cu wires, and also slightly different milling parameters than
those used in Fig. 5.3. Samples sam10b3p1 and sam10b3p3 [Fig. 5.4(a)] have 34 nm
thick Py and 41 nm thick Cu lines, with 80 seconds of ion milling with 10?4 Torr of
argon at 250 V, 100 V accelerating voltage, and 5 mA current. Sample sam12b1p2
[Fig. 5.4(b)] has 48 nm thick Py and 76 nm thick Cu lines, and the same ion milling
parameters. The fits to equation (3.29), which give spin relaxation lengths of 920nm
81
0 0.2 0.4 0.6 0.8 1 1.2
L (?m)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
R A
(m
?)
0 0.2 0.4 0.6 0.8 1 1.2
L (?m)
0
0.1
0.2
0.3
0.4
R A
(m
?)
(a) (b)
=960 nm?? =920 nm
Figure 5.4: Nonlocal resistance switching (RA) as a function of injector-detector
separation L for (a) the average of samples sam10b3p1 and sam10b3p3, and (b)
sample sam12b1p2, with transparent Py-Cu contacts at 4.2 K.
(ap =1.49?10?3) and 960nm (ap =9.2?10?4), are also shown. The extracted values
of ? (and ap) are also in agreement with the results from previous measurements
quoted above, but the scatter in the data still gives confidence intervals which are
too large. The scatter in the data is attributed to lack of reproducibility in the
contacts between Py and Cu due to differences in the size of the contact area,
unremoved resist, and incomplete oxide and surface contamination removal by the
ion mill. Efforts to increase sample reproducibility were not completely effective,
but the importance of using longer ion mill times and larger line thickness is evident
after comparing Figs. 5.3 and 5.4. For longer milling times and thicker Cu wires
the size of RA is larger.
In conclusion, it was shown that although it is in principle possible to ex-
tract the spin diffusion length from the length dependence of the spin-dependent
resistance, this procedure can have a large systematic error coming from the par-
ticularities of each of the Py-Cu contacts. The average value of the spin diffusion
length found from the fits is 927 nm (with ap =8.9?10?4) which is comparable to
82
values found in the literature3, but the confidence intervals are large, limiting the
credibility of the results. In the next section I will present an alternative method
for finding both the spin diffusion length and the degree of spin polarization with a
single measurement, evading the issue of contact reproducibility, and yielding more
accurate results.
5.2.2 Spin precession: Hanle effect
In this section I will present results of spin precession (Hanle effect) experiments,
from which the spin diffusion length and the spin polarization can be determined.
All the samples used for this experiments had Co-Al2O3-Cu junctions with contact
resistances varying between 9 ? and 1 k?, depending on the oxidation parameters.
A schematic of the experiment is shown in Fig. 5.5 (b). First the magnetic contacts
were aligned by applying an in-plane magnetic field. Then this in-plane field was
turned off andan out-of-planemagnetic fieldB? was applied to induce the precession
of the spins as the electrons diffused from the injector to the detector.
Figure 5.5 (a) shows some typical Hanle effect data, together with a fit to
the model presented in chapter 3 [equation 3.38]. This particular data is for sample
s sam11b2y1p2, which has a separation of L=425 nm between injector and detector,
at a temperature of 4.2 K. AsB? increases, the spin, which lies in plane, precesses by
a larger angle, and is no longer aligned with the detector, decreasing the magnitude
31000 nm (ap =6.9?10?4) for spin injection by Jedema et. al. [22], 700-1400 nm for energy
level spectroscopy by Petta et. al. [62], and 450 nm (ap =3.2?10?4) for GMR measurements by
Yang et. al. [58]. Conduction electron spin resonance (CESR) measurements give 130 times larger
spin relaxation times, but equally larger mean free times, so that the ratio of the two time scales
is similar (ap =9?10?4) [63, 64].
83
? 300 ? 200 ? 100 0 100 200 300
B?(mT)
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Nonlocal
resistance
(m
?)
??
? = 22.2 ps
?N = 546 nm
P1P2 = 5.5%
(b)
F2
F1
N 21N
T1 T2
I V
0000
0000
0000
0000
0000
1111
1111
1111
1111
1111
00
00
00
00
00
00
00
11
11
11
11
11
11
11L
0
0
1
1
0
0
1
1(a)
B
Figure 5.5: (a) Hanle effect measurement of sample s sam11b2y1p2 at 4.2 K for
aligned injector and detector separated by 425 nm; the nonlocal resistance is shown
asafunction oftheout-of-planemagneticfieldB?. Afitto thedatais alsoshown, to-
gether with the fitting parameters?=22.2 ps and?P1P2=5.5%. (b) Schematic of the
Hanle effect experiment. The nonlocal resistance, defined as RNL = VT2?N2/IT1?N1
for either aligned or anti-aligned magnetizations of injector and detector is measured
as a function of the out-of-plane magnetic field B?.
of the measured nonlocal resistance. For large enough magnetic fields, the spin
can rotate 180 degrees, and the nonlocal resistance can change from positive to
negative. However, as B? is increased, the effective magnetization of the injector
and detector begins to rotate and starts to have an out-of-planecomponent, affecting
the interpretation of the measurement. This is the main reason to avoid going to
higher magnetic fields. Figure 5.5(a) also shows the values of the fitting parameters
which are ?=22.2 ps and ?P1P2=5.5%. The spin diffusion length, calculated using
?N = ?D?, is ?N =546 nm. Although no independent information about P1 and
P2 can be obtained, their product is known, and if the contacts are assumed to have
similar characteristics, then it can be claimed that the current spin polarization
at the interface is 5.5%. Furthermore, even without any assumptions it can be
inferred that the current spin polarization at one of the interfaces is at least 5.5%,
84
the geometric mean of P1 and P2.
Figure 5.6 shows two complete sets of data (both aligned and anti-aligned
configurations of the magnetization of injector and detector) showing the Hanle
effect for two different separations between injector and detector at 4.2 K. Parts (a)
and (c) are for a separation of 867 nm, while (b) and (d) are for a separation of
1110 nm. The figures on the left are well fit by the same parameters, ?=25.3 ps and
?P
1P2=3.8%, while the ones onthe right are best fit by slightly different parameters,
?=26.0 ps and ?P1P2=4.3% for aligned, and ?=27.6 ps and ?P1P2=4.6% for anti-
aligned. However the difference between them is small. The values of the spin
current polarizations and the spin diffusion lengths are comparable to those shown
in Fig. 5.5(a). Besides the larger separation between injector and detector, the
main difference between these samples and that shown in Fig. 5.5 is that these were
oxidized at a pressure of 122 mTorr while the other one was oxidized at a pressure
of 410 mTorr.
However there are two important features that should be noticed from Fig.
5.6. The first one is that Fig. 5.6(b) is slightly asymmetric in field. This can be un-
derstood from the fact that the magnetization in injector and detector are sometimes
not exactly parallel4. The zero in-plane field state of the magnetic contacts is that
which is a local minimum of the energy stored in the magnetic system. This energy
is a function of the domain structure of each contact, which is sensitive to the shape
4In the extreme case in which injector and detector are perpendicular to each other, the Hanle
effect signal would be completely antisymmetric in field, since for positive fields precession tries
to align the spins with the detector, while for negative fields precession tries to align the spins
opposite to the detector.
85
? 300 ? 200 ? 100 0 100 200 300
B?(mT)
? 0.4
? 0.3
? 0.2
? 0.1
0
0.1
0.2
0.3
0.4
??? = 27.6 ps
?N = 649 nm
P1P2 = 4.6%
? 300 ? 200 ? 100 0 100 200 300
B?(mT)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
??
? = 26 ps
?N = 631 nm
P1P2 = 4.3%
? 300 ? 200 ? 100 0 100 200 300
B?(mT)
? 0.7
? 0.6
? 0.5
? 0.4
? 0.3
? 0.2 ??? = 25.3 ps
?N = 622 nm
P1P2 = 3.8%
? 0.1
0
0.1
0.2
0.3
0.4
? 300 ? 200 ? 100 0 100 200 300
B?(mT)
??
? = 25.3 ps
?N = 622 nm
P1P2 = 3.8%
(a) (b)
(c) (d)
Nonlocal Resistance (m   )
? Nonlocal Resistance (m   )
?
Figure 5.6: Hanle effect measurements and fits for two different injector-detector
separations (sample s sam11b2y3p2) at 4.2 K. (a) 867 nm, aligned injector and
detector (??). (b) 1110 nm, ??. (c) 867 nm, ??. (d) 1110 nm, ??.
anisotropy and to the crystalline structure of the material. The fabricated films
are polycrystalline so the crystalline structure plays a much smaller role than the
shape anisotropy in defining the domain structure of the contacts. In the fabricated
samples single domain structure throughout most of the ferromagnetic contacts is
expected due to the strong shape anisotropy and the reduced size, but small closure
domains at the edges are typical in rectangular shaped structures. This closure do-
mains can be affected by the local crystalline structure which is sample dependent,
and can also depend on the history of the magnet. Depending on the details of
the closure domains, the magnetization of the contact can be slightly rotated from
the main geometrical axis, therefore the magnetization of injector and detector can
sometimes be not completely parallel.
The second feature of importance is that the maxima of the data in parts (a)
86
and (b) have different magnitude than the corresponding minima in figures (c) and
(d). This means that there is a shift in the value of the nonlocal resistance which
can be positive or negative. This shift can be explained by a small leakage current
into the finite input impedance of the lock-in amplifier, and its magnitude and sign
depend on the values of the resistance of different sections of the sample, as well
as contact, and probe wiring resistances. Simulations using PSPICE (which will be
shown later in this chapter) show that the value of this shift is of the order of a few
tenths of a milliohm, which gives the correct order for the observed shifts in Fig.
5.6.
Finally, to test the linearity in bias current, Fig. 5.7 shows Hanle effect data
for currents of (a) 19.6 ?A and (b) 9.8 ?A at 4.2 K for sample s sam12p2 with an
injector-detector separation of 717 nm. The fitting parameters are almost identical
in both cases, confirming the linearity of the nonlocal resistance on applied bias
current. The spin diffusion length is similar but slightly larger than that measured
for other samples, but the surprising result is the very high value of the spin current
polarization P of 17.2%, compared with other fabricated samples which had P close
to 5%, and which is only one half of the value in bulk Cobalt of 35%. The main
difference between this and previous samples is that the oxidation pressure was 423
mTorr. Even though this pressure is close to the value of 410 mTorr quoted for
sample s sam11b2y1p2 and shown in Fig. 5.5 which had P of only 5.5 %, it was
seen experimentally that the contact resistance was very sensitive to the pressure
in the 400 mTorr range. This provides evidence that by controlling the oxidation
87
? 300 ? 200 ? 100 0 100 200 300
B?(mT)
1
2
3
4
5
6
7
8
Nonlocal
resistance
(m
?)
??
? = 23.28 ps
?N = 569 nm
P1P2 = 17.2%
(a)
? 200 ? 100 0 100 200
B?(mT)
0
2
4
6
8
Nonlocal
resistance
(m
?)
??
? = 23.32 ps
?N = 569 nm
P1P2 = 17.2%
(b)
Figure 5.7: Hanle effect measurement for sample s sam12p2 with injector-detector
separation of 717 nm at 4.2 K, for (a) 19.6 ?A bias and (b) 9.8 ?A bias. Note the
large value of the spin current polarization, about three times larger than for other
samples, but a comparable spin diffusion length.
pressure precisely in the 400 mTorr range, the spin current polarization can be
greatly enhanced.
In conclusion, the spin diffusion length of electrons in Copper was measured
to have an average value of 601 nm ? 36 nm (ap=6.2?10?4), which is comparable
to previous results from GMR measurements (450 nm, ap=3.2?10?4) [58], spin
injection (1000 nm, ap=6.9?10?4) [22], and energy level spectroscopy (700-1400
nm) [62]. These values are smaller than those obtained in the previous section
from the length dependence of the nonlocal resistance by roughly a factor of 1.5.
However, the ratio of the momentum and spin relaxation times ap is also smaller,
which means that the Hanle effect samples have a higher resistivity than the ones
measured before. Spin current polarizations were typically 5% but some samples
showed much larger values approaching 20%. The conditions under which samples
with very different values of P were fabricated are similar, which shows the extreme
sensitivity of the interfacial spin transport to processing details such as the oxidation
88
parameters, in particular the oxidation pressure.5.
5.2.3 Symmetric component of the nonlocal resistance
According to the model for spin transport presented in chapter 3, in particular
equations (3.31) and (3.34), the nonlocal resistance should have the same magnitude
for both aligned and anti-aligned magnetizations of injector and detector, but it
should be positive for aligned and negative for anti-aligned magnetizations. In this
section I will present data that shows that this is true at low temperatures, but
at higher temperatures there is an offset or symmetric component of the nonlocal
resistance which is also related to spin transport, and which can be explained by
different interface spin-flip scattering for spin ? and spin ? electrons at the detector
(F2) interface.
The temperature dependence of the nonlocal resistance was measured for the
two possible injector-detector configurations. First an in-plane magnetic field was
used to align injector and detector (??). Then the field was turned off and the
temperature was changed from room temperature down to 77 K and back up to
room temperature. The in-plane field was then turned on to align injector and
detector in opposite directions (??). The field was then turned off and the tem-
perature was again oscillated slowly between room temperature and 77 K. Due to
temperature instability close to liquid nitrogen temperature, only the data above
5Some factors that can affect spin transport at the interface are antiferromagnetic oxide for-
mation, magnetic fields produced by inhomogeneities in the magnetic material at the interface,
spin-flip scattering, varying oxide thickness, or even contact potentials that can affect the local
density of states of the metals. However, understanding this details is outside of the scope of this
work.
89
0 50 100 150 200 250 300
Temperature (K)
? 4
0
4
8
12
Nonlocal
resistance
(m
?)
Figure 5.8: Temperature dependence of the nonlocal resistance for aligned (??) and
anti-aligned (??) magnetizations of injector and detector for sample s sam11b2y1p2
with an injector-detector separation of 425 nm. Dashed lines show the 4.2 K values.
100K are shown. Figure 5.8 shows typical traces of the temperature dependence of
the nonlocal resistance for both configurations, with the dashed lines representing
the values measured at 4.2 K.
At 4.2 K the values of the nonlocal resistance for the two different configu-
rations have the same magnitude but different sign, as expected from the model.
However, at 115 K it is clear that this is no longer true and both values have shifted
in the positive direction. As the temperature is increased further, the offset increases
nonlinearly and at room temperature it is more than three times larger than the
low temperature value of the nonlocal resistance. For comparison, the temperature
dependence of the resistivity of both copper and cobalt was measured, as well as
the temperature dependence of the contact resistance. The results are shown in
90
? Co
(??
cm
)
22
24
26
? Cu
(??
cm
)
4
4.5
5
5.5
100 200 300
T (K)
R C
(?
)
12.9
13.3
13.7
(a)
(b)
(c)
Figure 5.9: Temperature dependence of (a) the resistivity of Co, (b) the resistivity
of Cu, and (c) the contact resistance.
Fig. 5.9. They all show a linear increase with temperature in the 100-300 K range,
which rules out many resistive effects as a possible origin for the nonlinear increase
in both R??NL and R??NL. In order to understand this better, the symmetric (?+?)
and anti-symmetric (?-?) components of the nonlocal resistance
RS,A = (R??NL ?R??NL)/2, (5.2)
are shown in Fig. 5.10. From this figure it is clear that the antisymmetric component
of the nonlocal resistance (RA), which is the only one that had been studied up
91
Figure 5.10: Symmetric and anti-symmetric components of the nonlocal resistance
as defined in equation (5.2), for the data shown in Fig. 5.8.
to now6, decreases linearly with increasing temperature in the 100 K to 300 K
range. However, the symmetric component, which had not been observed up to now,
increases nonlinearly in that temperature range. As opposed to the antisymmetric
component, the symmetric one is by definition that component which is independent
of the relative alignment of injector and detector. This indicates a possibly different
origin for the two signals, which does not mean that the symmetric component is
not related to spin-transport, and evidence will be given to support the claim that
6With Tunnelling Magneto Resistance (TMR) geometries only RA can be measured since even
in the absence of spin effects there is a large symmetric component or offset arising from resistive
effects.
92
it is in fact related.
The transport model described by equations (3.29), (3.31), and (3.34) cannot
explain the existence of RS since they predict a vanishing symmetric component of
the nonlocal resistance. However, if interface spin-flip scattering takes place, these
equations are no longer valid, and it is necessary to generalize them to include spin
non-conserving currents at the different interfaces. This was done in the last section
of chapter 3, where the relevant equations (3.40) and (3.41) were deduced. It was
shown that for certain kinds of spin-flip scattering the nonlocal resistance could have
an additional component which did not reverse its sign when the magnetization of
the injector was reversed, and which was due to the difference between P??2 and P??2 .
This would allow the existence of a symmetric component of the nonlocal resistance.
A detailed discussion follows.
Phenomenology and analysis of RA and RS
In the limit of large contact resistances the model discussed in chapter 3 predicted
that the nonlocal resistance would be given by [see equation (3.34)]
RNL = ? 12A
N
?N?exp(?L/?)P1P2, (5.3)
where P1 had the physical significance of being the spin current polarization at the
detector interface, while the ?+? or ?-? signs applied to aligned (??) or anti-aligned
(??) injector and detector. The product ?N?/2AN exp(?L/?) is related to the spin
transport in the nonmagnetic material therefore it should be the same for ?? and
??. Additionally, the value of P1, the spin current polarization at the injector, is
93
independent of the direction of magnetization of the detector, P1 ?P??1 ?P??1 . On
the other hand since the detector can be aligned or anti-aligned with the injector,
transport across the detector interface can be characterized by different values of
P2, namely P??2 for aligned and P??2 for anti-aligned. Therefore by using equation
(5.3), RS,A = (R??NL ?R??NL)/2 can be rewritten as
RS,A = 12A
N
?N?exp(?L/?)P1PS,A, (5.4)
with the definition
PS,A = (P??2 ?P??2 )/2, (5.5)
where the ?+? (?-?) sign refers to the symmetric (antisymmetric) component of P2.
It is possible to analyze the temperature dependence of equation (5.4) further as
follows. The term ??N? (the resistivity of copper) was measured in the 100 K -
300 K temperature range as shown in Fig. 5.9(b). The relaxation length ? (which
appears both as a multiplying factor and in the exponential term) can be related
to the spin relaxation time ? by using the diffusion relation ? = ?D?. According
to Elliot and Yafet [5, 4] and as discussed in chapter 2, the spin relaxation time
is proportional to the momentum relaxation time ?p. The proportionality constant
is temperature independent at high temperatures, where the dominant source of
both momentum and spin relaxation is phonons. By using equation (2.4) it is then
possible to relate ? to ?p by
94
? =
radicalBig
D?p/ap. (5.6)
The momentum relaxation time can be calculated in terms of the resistivity by using
the Drude model to give
?(?N) =
radicalBigg m?D
ne2ap?N, (5.7)
where m? is the effective mass of electrons in copper, n is the Cu electron density,
and e the proton charge. This means that, up to the constant ap, the temperature
dependence of ? can be obtained from the measured temperature dependence of
?N in the 100-300K temperature range. At low temperatures the value of ap can
be calculated from our previous measurements of the spin relaxation time and the
resistivity of copper at 4.2 K. However, since the scattering mechanisms responsible
for momentum and spin relaxation at high temperatures are different from those
at low temperatures, ap should be treated as a fitting parameter. Additionally, the
temperature dependence of the term ?P1?, the injector spin current polarization,
has the form P1 = P0(1??T3/2) [65],7 with the additional fitting parameter ?. The
value of ? is known for bulk but since the injected spin polarization is an interface
effect, values much larger than those for bulk are expected. The value of P0 (5.5%)
was measured by using Hanle effect in the previous section. It has been shown [65]
that the previously observed PA has the same temperature dependence as that of
7Spin waves in ferromagnetic materials are responsible for the decrease of the magnetization
as the temperature increases with the dependence M = M0(1 ??T3/2) [66]. If the spin current
polarization is assumed to be proportional to the magnetization, the temperature dependence of
M should be reflected in P.
95
the injector. Therefore the functional form of the temperature dependence of the
right hand side of RA in equation (5.4) is known up to two unspecified constants ap
and ?. This functional form, given explicitly by
RA = 14A
N
P20(1??T3/2)2?(?N)?N exp(?L/?(?N)), (5.8)
where ?(?N) is given by equation (5.6), can be used to fit the data for RA shown
in Fig. 5.10 using ap and ? as fitting parameters. The values obtained from the
fit are ap=6.6?10?4 which is 1.06 times larger than the value of ap measured at
4.2K and agrees very well with the results of Jedema et. al [22] (6.9?10?6), and
? =8.4?10?5K?3/2, which is, as expected, larger than in bulk [65]8.
Once the parameters ap and ? have been found from the fit, it is possible to
calculate PA and PS by inverting equation (5.4) since the temperature dependence
of all the additional terms is determined by the two fitting parameters. The result of
such calculation is shown in Fig. 5.11. The temperature dependence of PA andPS is
similar to that ofRA andRS due to the almost linear temperature dependence of the
additional factors in equation (5.4). Furthermore PA is fit well by the assumed form
P0(1??T3/2), which confirms the correct choice of fitting function. Additionally RS
can be fit by a Fermi function B/(1+expT0/T) with an activation temperature T0
of 1227K.
8The magnetization, and therefore the spin polarization, is more robust in bulk than at the
interfaces, where the ferromagnetic state is more sensitive to thermal effects.
96
Figure 5.11: Temperature dependence of PA and PS together with fits to P0(1 ?
?T3/2) (with P0=5.5% and ?=8.4?10?5K?3/2) and a Fermi function B/(1 +
expT0/T) with an activation temperature of T0=1227K.
Theory and physical interpretation of PA and PS
In the absence of spin-flip scattering at the interfaces, both the charge and spin
current are conserved. In this case the boundary conditions given by equation
(3.15) and the corresponding schematic shown in Fig. 3.2 describe transport at
each interface. However, it was shown in chapter 3 that in the case of interfacial
spin-flip scattering the correct boundary conditions were given by equations (3.39)
and the corresponding schematics shown in Fig. 3.11. By using this new set of
boundary conditions it was shown that the nonlocal resistance was given by the same
expression as the one used in the absence of interfacial spin-flip scattering, namely
equation (5.3), with redefined constants Pi (i = 1,2). The general expressions for
the Pi were not very enlightening [see equations (3.41)]. However, assuming that
97
the spin-conserving interfacial resistances (R?) depend only on the magnetization
of the ferromagnet, and that the spin-flipping interfacial resistances depend on the
nonequilibrium magnetization in the nonmagnetic metal, it is possible to simplify
equations (3.41) to obtain9
P1 = (1/R1? ?1/R1?)(1/R
1? + 1/R1?) + (1/R?1? + 1/R?1?)
P??2 = (1/R2? ?1/R2?) + (1/R
?
2? ?1/R
?
2?)
(1/R2? + 1/R2?) + (1/R?2? + 1/R?2?)
P??2 = ?(1/R2? ?1/R2?) + (1/R
?
2? ?1/R
?
2?)
(1/R2? + 1/R2?) + (1/R?2? + 1/R?2?) . (5.9)
Therefore the symmetric and antisymmetric combinations PS,A = (P??2 ?P??2 )/2 are
given explicitly by
PS = (1/R
?
2? ?1/R
?
2?)
(1/R2? + 1/R2?) + (1/R?2? + 1/R?2?)
PA = (1/R2? ?1/R2?)(1/R
2? + 1/R2?) + (1/R?2? + 1/R?2?)
. (5.10)
The denominator of both of these equations is proportional to the inverse of the
contact resistance, which was measured (see Fig. 5.9) and found to be almost con-
stant in the 100-300K temperature range. This means that it is the temperature
behavior of the numerators of both of equations (5.10) which causes the particular
temperature dependence of PS and PA shown in Fig. 5.11. The physical significance
of the quantities PA and PS can now be understood from equation (5.10). PA gives
the ratio of spin-polarized to total transport from Cu into F2 (the detector). This
9The following P?s have been defined so that the ? sign in front of equation (5.3) should not
be used, since it has already been included into the new definition of P2.
98
new interpretation clarifies that at low temperatures P2 gives a measure of the spin
polarized transport that differs from the spin current polarization, which diverges
due to the absence of charge current. In addition, PS provides information on the
ratio of differential spin-flipping to total transport from Cu into the detector. In
other words, it compares the spin-flip transport of the two spin channels, normalized
by the total transport across the N?F2 interface. Therefore as the temperature in-
creases up to 300 K, the amount of spin polarized transport from Cu to the detector
decreases to about 3%, while the difference in spin-flipping transport of spin ? and
spin ? increases to about 30%. This can be expected since it has been shown that
in some ferromagnetic materials the mean free path of majority electrons is more
than 80% larger than the mean free path of the minority electrons [67].
In conclusion, the existence of spin-flip scattering at the interface can explain
the symmetric component of the nonlocal resistance observed in the measurements.
Since interfacial spin-flip scattering increases with temperature, it is expected that
the symmetric component of the nonlocal resistance also increases with temperature,
as confirmed by the measurements. The new observed component of the nonlocal
resistance, RS, is fit well by a Fermi function, typical for thermally activated mech-
anisms, which further confirms the proposed explanation of enhanced interfacial
spin-flip.
In order to check for possible artifacts as the origin of RS, its dependence
on the separation between injector and detector was measured. Temperature and
length dependence measurements of nonmagnetic control samples were also done.
99
Additionally, tests were performed to rule out thermoelectric effects produced by lo-
cal sample heating, geometric electrostatic effects, current leakage through the input
impedance of the lock-in amplifiers, resistive leakage in the probe wiring, capacitive
leakage, DC and second harmonic leakage of signal into the lock-in amplifiers, and
out of phase detection due to dephasing. In the following sections I will describe
these measurements and tests and give their results.
5.2.4 Length dependence of RS
Figure 5.12 shows the symmetric component of the nonlocal resistance as a function
of the separation L between injector and detector for two different samples, one (a)
with transparent Py-Cu contacts and seven working magnetic contacts, the other
(b) with transparent Co-Cu contacts and eight working magnetic contacts. Mea-
surements were taken at room temperature. Each of the different symbols in Fig.
5.12 represents a different injector, and for a given injector the signal is measured at
detectors placed at different distances from the injector. For many of the measure-
ments there are intermediate magnetic contacts between injector and detector. As
L increases the value of RS decreases, as expected for a signal that is related to spin
injection. Furthermore, the length dependence of the data is fit well by the model
discussed in chapter 3 for transparent interfaces which was used to fit the decay
of the antisymmetric component of the nonlocal resistance RA. The fits give decay
lengths?S of 875 nm and 943 nm for the samples shown in (a) and (b) respectively10.
The values of ?S are close to the average value of the spin diffusion length measured
10Fits to the data for a single injector gave similar values of the fitting parameters as those
obtained by fits to all of the data.
100
0
5
10
15
20
25
R S
(m
?)
0 1000 2000 3000 4000 5000
L (nm)
?S=875 nm
0 1000 2000 3000 4000 5000
L (nm)
0
1
2
3
4
5
6
R S
(m
?) ?S=943 nm
(b)
(a)
Figure 5.12: Injector-detector separation (L) dependence of the symmetric part
of the nonlocal resistance RS for transparent samples with (a) Py-Cu interfaces
(s sam10b3) and (b) Co-Cu interfaces (s sam10b1p1) at room temperature.
101
0 1000 2000 3000 4000 5000
L (nm)
0
2
4
6
8
10
R S
(m
?)
Py,Cu
Cu,Py
Cu,Cu
Py,Py
Figure 5.13: Injector-detector separation (L) dependence of the symmetric part of
the nonlocal resistance RS for transparent samples with either Py-Cu or Cu-Cu
interfaces. Measurements for samples with Py injector and detector (Py,Py), Py
injector and Cu detector (Py,Cu), Cu injector and Py detector (Cu,Py), and Cu
injector and detector (Cu,Cu) are shown.
in section 5.2.1 by studying the dependence of RA on L (927 nm), and are roughly
50% larger than the average value measured using Hanle effect (601 nm). The fact
that length dependence measurements of both RS and RA give similar decay lengths
provides evidence that both components are spin-related. Further evidence that RS
is related to spin is provided by Fig. 5.13, which shows measurements similar to those
shown in Fig. 5.12, but with injector and detector fabricated of different materials.
Only the measurements on samples with both magnetic injector and detector show
length dependence of RS, while the remaining measurements where either or both
of the magnetic contacts were replaced for Cu showed no length dependence.
102
100 150 200 250 300
Temperature (K)
0
5
10
15
20
R S
(m
?)
Co injector and detector
Ti injector and detector
Figure 5.14: Comparison of the temperature dependence of RS for a sample with
magnetic injector and detector, and a control sample with nonmagnetic Ti elec-
trodes.
5.2.5 Temperature dependence of RS for a control sample
Control samples were fabricated by replacing both the magnetic injector and detec-
tor by titanium (Ti). The typical resistance of the Ti-Cu contact was about 80?,
about 6 times larger than the contact resistance of many of the Co-Cu samples
that showed a nonlinear RS. With the same setup that was used to measure the
mentioned Co-Cu samples, temperature dependence measurements of the symmetric
nonlocal resistance RS were performed on the Ti-Cu control sample. A comparison
of the results of the two measurements is shown in Fig. 5.14. The symmetric nonlocal
voltageVS is shown instead of the nonlocal resistance RS. The control sample shows
a very small temperature dependence, two orders of magnitude smaller than that of
103
the Co-Cu sample, confirming that the signal is related to spin. As will be argued
later, it also shows that geometric effects are not an issue in our measurements.
5.2.6 Heating and thermoelectric effect
If two different metals which are in contact with each other are at different temper-
atures, an electric potential difference appears between them [Fig. 5.15 (a)]. This is
known as thermoelectric effect, thermopower, or Seebeck effect. The thermoelectric
voltage between points A and B in Fig. 5.15 (a) is given by
VA ?VB =
integraldisplay B
A
S(x)?T(x)dx, (5.11)
where S(x) is the Seebeck coefficient, which depends on position since it is different
for different metals, and T(X) is the temperature throughout the two metals.
The possibility of heating effects appearing as an additional voltage in the
measured nonlocal signal can be understood as follows. Figure 5.15 (b) shows a
schematic of a typical device together with the path of the current and the voltage
probe locations. The gray regions depict ferromagnetic contacts while the white
ones represent the nonmagnetic metal wires. Since the resistivity of the magnetic
materials used is larger than that of the normal metal, most of the Joule heating
(I2R) will occur in the path of the current before point A. The temperature should
therefore be highest in this region and decrease with the distance from point A.
Points D and E are far away from the heated area and should therefore be at the
same temperature. However points B and C are close to A so they can be at a
different temperature, which depends on the details of heat diffusion in the sample.
104
F1
T1
x
2
A T
Metal 1 Metal 2
F2
B
0000
0000
0000
0000
0000
1111
1111
1111
1111
1111
0
0
0
0
0
0
0
1
1
1
1
1
1
1V
I
N
(a) (b)
E
A
C
D
B
0
0
1
1
Figure 5.15: (a) Schematic of Seebeck effect with two dissimilar metals. (b)
Schematic of a typical sample to illustrate the possibility of measuring heating-
related voltages.
If the segment between B and C was also a normal metal, then from equation 5.11
it is clear that VB ?VE = VB ?VD since the Seebeck coefficient would be position
independent. But since segment B-C is made of a magnetic material with a different
Seebeck coefficient than that of copper, then VD ?VE negationslash= 0 and a heating-related
voltage might appear between the voltage probes. The thermal mass of the system is
small so the thermal response is fast. Simplified calculations show that the thermal
response time is much shorter than the period of the AC drive current, so the
temperature and therefore the Seebeck voltage have the same time dependence as
the heating, proportional to I2. Assuming that the current drive has both DC and
AC components, the heating-related voltage that appears between points D and E
is given by
VS = VD ?VE ? (IDC +IAC(t))2
? (I2DC +I2AC/2) + 2IDCIAC sin?t?I2AC cos(2?t)/2. (5.12)
105
The first term in the second line of equation 5.12 appears at DC, the second
one appears at the fundamental, and the last one shows up at the second harmonic
and is 270 degrees out of phase. In principle lock-in measurements are only sensitive
to the fundamental component (and to odd harmonics), so the only term that could
be measured is 2IDCIAC sin?t. However, to make sure none of the other terms
could be measured, the DC and 2? rejection of the lock-in was tested, confirming
that this signals can only contribute to at most 1 nv lock-in output, which is two
orders of magnitude smaller than the observed symmetric component of the nonlocal
resistance RS, and can therefore be completely disregarded when measuring at the
fundamental frequency. To study the remaining component of the heating signal,
differential resistance measurements were performed. A relatively small AC current
with a rms amplitude of either 672 nA or 1344 nA was added to a DC current which
was changed between -30 and 30 ?A, and the nonlocal voltage was measured at
the AC frequency of the current. The result of the measurement is the differential
resistance
R(IDC) = ?V?I
vextendsinglevextendsingle
vextendsinglevextendsingleI
DC (5.13)
times IAC. Figure 5.16 shows the measurement results together with linear fits. The
triangles (squares) and solid (dashed) line are for an AC bias current of 672 nA (1344
nA). The data are fit well by a line with non-vanishing slope and intercept, which
means that the I-V curve has linear plus quadratic behavior. The quadratic term in
the I-V curve (the linear term inIDC in the fits) canbe attributed to heating together
106
DC
DC
? 30 ? 20 ? 10 0 10 20 30
IDC (?A)
14
16
18
20
22
24
26
28
G
(m
?)
20.8 m??0.235 m?/?A I
21.0 m??0.222 m?/?A I
R
Figure 5.16: Differential resistance R as a function of the DC bias current for AC
bias currents of 672 nA (triangles) and 1344 nA (squares), together with linear fits.
The solid (dashed) line is the result of the fit to the 672 nA (1344 nA) data.
with thermoelectric effects, but the linear term in the I-V curve (the constant term
in the fits) cannot. Furthermore, the heating effect is much smaller than the other
part of the measured signal for small currents, and even for currents of 20?A it is less
than 11% of the measured signal. This is the first piece of evidence that shows that
heating cannot explain the results of the measurements, since even though it can be
present, it does not give the main contribution to the measured signal. According
to equation 5.12, heating can be observed at the second harmonic. Since no spin-
related signal is expected at that frequency, measurements at the second harmonic
should only give a heating-related signal. Lock-inmeasurements (aswell as spectrum
analyzer measurements) were made to test this, and the result, ?0.243m?/?A I
107
120 140 160 180 200 220 240 260
T (K)
0
1
2
3
4
R S
(m
?) 10 V
5 V
2 V
1 V
Figure 5.17: Temperature dependence of RS for four different bias conditions as
indicated in the text.
agrees well with that from the differential resistance measurements, confirming the
presence of small heating-effects that cannot account for the observed symmetric
component of the nonlocal resistance.
As a final test of the effects of heating, the temperature dependence of RS
was measured for four different bias conditions, as shown in Fig. 5.17. The bias
conditions are the following:
Bias Label IAC (?A) IDC (?A)
1 V 9.8 4.9
2 V 19.6 9.8
5 V 49 24.5
10 V 98 49
Table 5.1: Bias parameters. The first column contains the applied voltage bias, the
second column the AC current bias, and the last one the DC current bias.
108
According to equation (5.12) any heating component observed at the fun-
damental frequency should be proportional to the product of IDC and IAC. This
means that any heating-related component of the nonlocal resistance should be pro-
portional to the DC current. However, Fig. 5.17 shows that for different DC biases
the curves are almost exactly the same. For the bias conditions 1 V and 2 V the
curves are indistinguishable. For 5 V, the curve has almost exactly the same shape
but is shifted towards negative values by a small amount, and for 10 V the shift is
larger and the shape of the curve is slightly different at the highest temperatures
but has not changed significantly. What this means is that only at high biases it
is possible to see heating effects, and they appear as shifts in the baseline that are
almost temperature independent. Furthermore, significant heating occurs only for
large AC and DC biases, and since in the typical measurement setup no DC cur-
rent is applied, heating and thermoelectric effects cannot account for the observed
symmetric component of the nonlocal resistance and its temperature dependence.
Finally, non-magnetic samples with Ti-Cu and AuGe-Cu interfaces were fabri-
cated and measured under the same conditions as other magnetic Py-Cu and Co-Cu
samples. Even though the contact resistances for the Ti-Cu samples were around
76 ?, larger than the contact resistance of any other sample for which RS was mea-
sured, the measured nonlocal voltage was less than 10 nV for all the Ti-Cu samples
and was almost temperature independent, supporting the idea that RS comes from
spin injection and detection rather than from heating and thermoelectric effects.
AuGe-Cu and Cu-Cu samples also showed null results.
109
5.2.7 Geometric effects
Another important issue that must be taken into account when designing spin valve
devices, is theeffect of thegeometry andelectrostatics inthemeasured signal. Figure
5.18 shows a simplified two dimensional picture of the electric field lines (or current
lines) and equipotential lines for a cross geometry when the current flows between
adjacent arms. The equipotential lines extend a certain distance lg to the bottom
andleft armsof thecross, even though thenet current flowing throughthis sections is
zero. This means that there is a nonzero potential difference between points A (close
to the cross) and B (far away from it). Therefore to avoid additional voltages in
spin valve measurements, the characteristic decay length for this equipotential lines,
lg, should be shorter than the separation between injector and detector. The value
of lg is related to the width of the arms of the cross. Detailed calculations for our
particular structures were not made, but according to the calculations of Takahashi
et. al. [26], the width of the wires was small enough compared to the separation
between injector and detector such that geometric effects were not important. A
supporting argument to disregard this effect is the fact that the resistivity of Co
and Cu and the contact resistance depend linearly on temperature as shown in
Fig. 5.9, and any electrostatic voltage should be linear in this variables, therefore
no nonlinear temperature dependence is expected for this effect, contrary to the
observed RS which is nonlinear in temperature.
More evidence that the observed nonlocal resistance does not come from elec-
trostatic effects is given by the fact that the Ti-Cu samples that were fabricated
110
Equipotential
       lines
Current LinesI
IA
B
C
Figure 5.18: Two dimensional schematic of the current lines and equipotential lines
in a cross geometry when the current flows between adjacent arms.
with contact resistances of about 76 ? gave null results.
As a side comment, it is important to note that measurements of the contact
resistance can contain a component from the mentioned electrostatic effect. To
illustrate this point, measurements of the contact resistance in Py-Cu samples were
done by injecting current as shown in Fig. 5.18 and measuring the voltage difference
between points C and B. The measurements gave negative values for the contact
resistance, that were on the order of -100 m?. The contact resistance must be
positive and in the case of transparent samples it must be smaller than hundreds of
m?. This apparent contradiction is resolved by looking at the equipotential lines
between points C and B in Fig. 5.18, and realizing that the equipotential line closest
to C is at a lower voltage than that closest to B, so indeed a negative potential drop
111
exists apart from any real contact potential. Therefore electrostatic effects can be
of importance for some of the measurements, but their contribution to the nonlocal
resistance is negligible.
5.2.8 Current leakage
The four types of current leakage that can occur in the experiment are the following.
First, in the probe wiring; second, through the surface of the samples or through
the insulating SiO2 layer to the doped Si; third, capacitive leakage due to wiring
capacitances and the input capacitance of the lock-in amplifiers; and fourth, resistive
leakage due to the finite input impedance of the lock-in amplifiers. Current leakage
in the probe wiring, the sample holder, and the sample, were checked with a high
input impedance multi-meter 11. All of the measured resistances were larger than 1
G?. The stray capacitance of the wiring (wire to wire and wire to ground) at the
operating frequency is close to 100 M?, so its contribution is as large as that from the
leakage through the input impedance of the lock-in amplifiers. A circuit model of the
device and probe is needed in order to understand the effect of these current leakages
in the measured signal. This is presented in Fig. 5.19(b), together with a sample
schematic [Fig. 5.19(a)]. The numbered capacitances (120 pF and 160 pF) are stray
capacitances from the wiring, while the ones labelled A and B (30 pF) are the input
capacitances of the lock-in amplifier. For example C1?3 is the capacitance between
lead 1 and lead 3, while CA is the input capacitance of the A channel of the lock-
in amplifier. The resistors labelled RC1 and RC2 represent the contact resistances
11HP digital multi-meter.
112
of the F1-N and F2-N interfaces, R1-2 represents the resistance of the copper line
between injector and detector, and RA and RB are the input impedances of the
two channels of the lock-in amplifier. The rest of the numbered resistors represent
the resistance from either the injector interface or the detector interface to the
BNC connectors in the probe, through each of the different leads numbered in Fig.
5.19(a). For example R3 is the resistance from the F1-N interface to BNC 3, which
is connected to lead 3, while R6 is the resistance from the F2-N interface to BNC 6,
connected to lead 6. The two arrows show the placement of control leads used to
measure the difference in voltage between the A and B input channels of the lock-in
amplifier. The result of PSPICE simulations on this and similar circuits showed
that, depending on the particular values of the resistances and for bias currents of
about 20 ?A, there was an in phase component of the nonlocal resistance due to
leakage that was smaller than 15 nV and that could be either positive or negative,
together with an out of phase component that could be larger (reaching 100 nV for
some of the samples) and that could also have positive or negative sign. The out of
phase component is of no consequence for the measurements since the lock-in only
detects signals that are in phase with the drive. The small value of the calculated
in phase signal can explain the small shifts of the nonlocal resistance that were
sometimes observed at low temperatures (for example in Fig. 5.6), but it cannot
account for the large symmetric component that appeared at higher temperatures
or for its temperature dependence.
113
5
43
6
1 2
(a)
(b)
00000
00000
00000
00000
11111
11111
11111
11111
00
00
00
00
00
00
11
11
11
11
11
11
I V
00
00
11
11
Figure 5.19: (a) Schematic of the sample with numbered leads. (b) Circuit model
schematic of sample plus wiring. The numbers in the labels of the components are
those illustrated in (a). Resistances are in units of Ohms and capacitances in units
of Farads, with the additional modifiers pico (p), kilo (k), and giga (g).
114
5.2.9 Lock-in tests
To check the operation of the lock-in amplifiers that were used in the experiments,
three different tests were performed. The first one was to check the stability of
the phasing, the second was to check the DC rejection, and the third was to check
the second harmonic rejection. The first check was performed by phasing the lock-
in with a 100 mV bias at low sensitivity as described in the operation manual
of the PAR-124 Lock-in Amplifier, and then measuring the in phase and out of
phase voltage components at a small resistor at a high sensitivity (100 nV). The
appearance of an out of phase signal would mean incorrect phasing in the high
sensitivity settings. However, for an in phase signal of 108 nV, the out of phase
component was 1.9 nV, within the noise of the measurements, so no dephasing at
high sensitivities was detected. For the second test, DC rejection, DC voltages
between -100 mV and 100 mV were applied directly to the inputs of channels A and
B of the lock-in, and the in phase component was monitored at a sensitivity of 100
nV. The measured voltages were between -2.6 nV and 2 nV, also within the noise of
the measurements. Finally, the second harmonic rejection was tested by injecting a
voltage signal at twice the fundamental into either channel A or B and monitoring
the output of the lock-in. For voltages larger than 10 ?V the lock-in overloaded,
but for voltages as high as 9 ?V the output was less than 1 nV. Therefore all the
tests show correct operation of the lock-in.
115
Chapter 6
Spin injection and magnetic force microscopy detection I: Introduction
The previous section reported on transport measurements of the spin relaxation
length and the spin current polarization in Py-Cu and Co-Cu spin valves. The
interpretation of these measurements required fitting the data to a model based on
the diffusion equation, which assumed different electrochemical potentials for each
spin channel, and also a particular set of boundary conditions at the F-N interfaces.
Since no microscopic understanding of interfacial spin transport is available, an
alternative experiment which can directly give the spin relaxation length and the
spin current polarization without requiring any spin transport model is useful. This
and the following chapters will describe such an experiment, based on the direct
magnetic force microscope (MFM) detection of the magnetic field produced by the
excess magnetization in a Cu wire. This excess magnetization is generated by the
injection of spin-polarized electrons from a neighboring ferromagnetic contact.
In this chapter the operation and simplest theoretical model of a scanning
probe microscope (SPM) such as a magnetic force microscope (MFM), electric force
microscope (EFM), or atomic force microscope (AFM), will be shown. The pro-
cedure for measuring a non-equilibrium spin polarization in a nonmagnetic metal
116
using MFM will be discussed, together with some of the improvements that were
made to the cantilever and the operation mode in order to enhance the resolution
and sensitivity. The following chapter will discuss the actual experimental setup
and the results.
6.1 Scanning probe microscope operation and model
A schematic of a magnetic force microscope (or equivalently an atomic or electric
force microscope) is shown in Fig. 6.1. By means of a drive piezoelectric, a can-
tilever with a magnetized tip1 is made to oscillate in the z direction at its resonant
frequency?0 =
radicalBig
k/m? wherek is the cantilever spring constant andm? the effective
cantilever mass2. A semiconductor laser is used to shine light on the back surface
of the cantilever. By monitoring the difference in light intensity between the upper
and lower sections of a four segment photodetector, normalized by the total light
intensity, the deflection of the reflected beam and therefore of the cantilever can be
found. The deflection information goes to a controller and computer which send a
feedback signal to both the Z drive and Z piezoelectrics to control the cantilever
vibration and to adjust the cantilever?s vertical position. The sample is mounted
on X, Y, and Z piezoelectrics which control the fine motion from nanometers to
tens of microns. Coarse movement along any of the three axis can be made using a
mechanical stage.
The MFM is operated in the tapping mode, in which the cantilever encounters
1The magnetic metal covered tip is necessary only for MFM. For EFM any metal can be used,
but nonmagnetic ones are preferred, and for AFM no metallic coating of the tip is necessary.
2Which is equal to one fourth of the mass of the cantilever.
117
Controller
Computer
Deflection
Z Drive Piezoelectric 
Laser
Cantilever
Tip
Sample
X,Y,Z Piezoelectric 
X,Y,Z Mechanical Stage
X
Y
Z
Photodetector
Figure 6.1: Schematic of a scanning probe microscope (MFM, EFM, or AFM) with
a fixed vibrating cantilever and movable sample stage.
the short range repulsive part of the Van del Waals interaction with the sample. This
decreases both tip and sample damage by avoiding shear forces. To get topography
information of the sample the Z piezoelectric is used to move the stage up until
the cantilever oscillation amplitude changes, due to Van del Waals repulsion, to a
specified setpoint. The cantilever is then said to be engaged to the sample. The
X,Y piezoelectrics are then used to laterally move the sample. Any change in the
topography of the sample changes the force between the sample and the cantilever,
changing the deflection. The controller senses this change and modifies the vertical
118
position of the cantilever by changing the voltage at the Z piezoelectric to keep
the deflection at the specified setpoint. This feedback system avoids crashing the
sample into the tip, and by monitoring the feedback signal (i.e. the voltage control
sent to the Z piezoelectric) a calibrated image of the topography of the sample can
be generated since the voltage-displacement relation of the X,Y,Z piezoelectrics is
known.
For the Digital Instruments Multi-Mode SPM used in the experiment, topog-
raphy and magnetic force detection are interleaved, that is, line scans of topography
and line scans of magnetic force detection are taken one after the other. First, trace
(increasing X) and retrace (decreasing X) topography lines are acquired. The can-
tilever is then raised to an average baseline at a specified height above the average
height of the sample to do the magnetic force detection. During magnetic force
detection the feedback loop is disconnected and the change in the phase of the os-
cillating cantilever is monitored by the computer. Again trace and retrace lines are
acquired, the cantilever is brought down to the original Z position, and the system
is ready to acquire the next topography line.
The simplest model for the operation of an MFM (EFM or AFM) treats the
cantilever as a forced, damped, harmonic oscillator with resonant frequency ?0 and
quality factor Q. Therefore the equation of motion for the vertical position of the
tip Z is given by
1
?20
?Z + 1
Q?0
?Z +Z = 1
k
bracketleftBigg
F0 exp(i?0t) +F|Z0 + ?F?Z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingleZ
0
(Z?Z0)
bracketrightBigg
, (6.1)
119
where F0 is the driving force from the Z drive piezoelectric, F = F(x,y,z) is an
external time independent force acting on the tip of the cantilever (either from
magnetic, electric, or Van der Waals interaction with the sample). This force is
assumed to be much smaller than the drive so that it can be expanded in a first
order Taylor series about the equilibrium position of the tip Z0 in the absence of
any force. The solution to equation (6.1) is given by
Z(t) = 1k
bracketleftBigg
F ? ?F?ZZ0
bracketrightBigg
Z0
+ QF0k
bracketleftBigg
i? Qk ?F?Z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingleZ
0
bracketrightBigg?1
exp(i?0t). (6.2)
The first term on the right hand side only introduces a shift in the equilibrium
position of the tip and is time independent. Phase sensitive detection averages this
term to zero. The second one, however, has harmonic dependence with phase given
by
? = ??2 ?arctan
bracketleftBiggQ
k
?F
?Z
bracketrightBigg
Z0
, (6.3)
which is the actual measured quantity in MFM. Since the force is small compared
with the drive, the phase shift introduced by the external force is small so it can be
approximated by
?? = ?Qk ?F?Z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingleZ
0
. (6.4)
120
6.2 Spin injection and MFM detection
It was shown in chapter 3 that when a current is driven from a ferromagnetic to
a nonmagnetic metal the current is spin-polarized and the spin, or equivalently
the magnetization, decreases exponentially away from the point of injection with a
characteristic length given by the spin relaxation ? of the electrons in the nonmag-
netic material. Detection of such a non-equilibrium magnetization can in principle
be done by taking advantage of the high resolution and high sensitivity of MFM.
The advantage of MFM detection is that the magnetization of the spin-polarized
electrons is detected by directly measuring its effect on a magnetized oscillating can-
tilever, so no transport model is necessary to interpret the results. Additionally, the
length dependence of the magnetization can be studied for a fixed injector, avoiding
the contact reproducibility problems found during the length dependence measure-
ments of the spin signal in the previous chapter. Transport measurements of the
length dependence of the spin related resistance required using different contacts for
injection and detection, each of which could have different interfacial spin current
polarization P, producing systematic errors in the measurement of ?. This is solved
with MFM detection since in MFM measurements a fixed magnetic electrode is used
for spin injection, and detection is done by scanning at different separations from
the injection point. Finally, the spin current polarization P of the injector can be
found independently, which was not possible with transport measurements where
only the product P1P2 could be found.
A very important issue that has to be addressed is the back-action of the stray
121
magnetic field of the cantilever on the electrons. Measurements [68] have shown that
the stray field from a typical cantilever at a distance of 250nm from the tip is about
4mT. It was shown in chapter 5 that in our typical devices electrons precess by an
average angle of ?/2 in a distance of the order of 900nm when a field of 300mT is
applied. Therefore the stray field produced by the cantilever is too small to produce
any significant electron spin dephasing in our Cu wires, so that the back-action of
the cantilever can be ignored.
Aschematic of theconfiguration forspin injection andMFM detection is shown
in Fig. 6.2. An oscillating cantilever with a magnetized tip (shown in light gray)
scans the surface of the sample in the x-y plane. Current is driven along the Cu
wire (shown in white) between the top (T) and bottom (B) leads. Close to y = 0 a
section of the Cu wire has been replaced by Py (shown in dark gray), so that all the
I
T
B
x
yz
Figure 6.2: Schematic for spin injection and MFM detection. A current driven along
the copper wire (white) is forced to enter a ferromagnetic section (black)of the wire,
as a cantilever with a magnetized tip scans close to the ferromagnetic-nonmagnetic
metal interface.
122
current goes from Cu to Py and back again to Cu. If the length of the wire section
that was replaced is longer than the spin relaxation length of electrons in Py (less
than ten nanometers according to Dubois et. al. [59] and Steenwyk et. al [61]), spin
transport across one of the Cu-Py interfaces does not depend on the spin transport
across the other Cu-Py interface. Assuming that the magnetic material has its
magnetization in-plane and pointing in the x direction (this direction will be called
spin up), electrons that enter the Py travelling from B to T have mostly spin up
since the density of states in Py for spin up is larger than for spin down. Therefore
an excess population of electrons with spin down is left in the bottom Cu section
close to the interface. The injected electrons lose their spin information within a
distance ?F, relaxing to the average spin polarization value in Py. Meanwhile on
the top Py-Cu interface mostly electrons with spin up are injected into Cu, so an
excess population of electrons with spin up will appear on the top Cu section close
to the magnetic contact. The electron density n? for spin ? can be easily calculated
from the electrochemical potential ?? by using the free electron model. In the limit
of small deviations from equilibrium, ??? ?EF where EF is the Fermi energy,
n?(y) = n0? + m
?
?h2?2kF????(y), (6.5)
where n0? is the equilibrium electron density, m? the effective mass, and kF? is
the Fermi wave vector. Furthermore, since the excess magnetization is given by
?M = ?B(n??n?) where ?B is the Bohr magneton, it is possible to write in general
123
?M(y) = m
??B
?h2?2 (kF????(y)?kF????(y)), (6.6)
or for the case of nonmagnetic metals where ??? = ???? = ?? and kF? = kF? = kF,
?M(y) = 2m
??B
?h2?2 kF??(y). (6.7)
By solving the equations for the electrochemical potential presented in chapter 3 for
each F-N interface, and using equations (6.5) to (6.7), the excess magnetization as a
function ofy can be calculated. For transparent F-N interfaces the calculated excess
magnetization (in the x direction) for a 1.2 ?m wide, 50 nm thick wire carrying a
current of 9.1 mA is shown in Fig. 6.3 for realistic values of the resistivity and spin
relaxation lengths of both metals. The qualitative features described previously
such as the extra spin up concentration for y > 0, extra spin down concentration
for y< 0, and independence of spin transport at both interfaces due to the fast spin
relaxation in the ferromagnet, are evident.
Detection of the excess magnetization is done by the oscillating magnetized
cantilever shown in Fig. 6.2. Since the cantilever has a tip magnetized in the z di-
rection, the interaction between the tip and sample magnetizations can be described
by the potential energy [69]
U = ?
integraldisplay
vectorM ? vectorB?MdV, (6.8)
where the term on the right hand side represents the energy stored in the magnetic
moment of the cantilever due to the stray field produced by the sample magnetiza-
124
? 0.4 ? 0.2 0 0.2 0.4
Position (?m)
? 4
? 2
0
2
4
?M
(A
/m
)
Position (y) (  m)?
Figure 6.3: Calculated excess magnetization ?M (in the x direction) as a function of
y for a 1.2 ?m wide, 50 nm thick wire carrying a current of 9.1 mA. Realistic values
of the resistivity and spin relaxation lengths of both metals have been assumed for
the calculation.
tion. Letting (?x?, ?y?, ?z?) be a reference frame mounted on the apex of the cantilever
tip, with each primed axis parallel to the corresponding unprimed axis of the fixed
reference frame on the sample (shown in Fig. 6.2), equation (6.8) can be written
more precisely as
U(vectorr) = ?
integraldisplay
vectorM(vectorr?)? vectorB?M(vectorr+vectorr?)d3r?, (6.9)
where vectorB?M(vectorr) is the magnetic field produced by the excess magnetization of the
device at a pointvectorrin the unprimed reference frame, while vectorM(vectorr?) is the magnetization
of the cantilever tip at point vectorr? in the primed reference frame. The energy is a
125
function of the relative position between the cantilever tip and the device. Since
the device is assumed to be fixed, the force on the cantilever produced by the
magnetic coupling with the sample is simply the negative of the gradient of the
potential energy. Furthermore, since the cantilever is extremely rigid in both x and
y directions its only degree of freedom is in the z direction so the magnetic driving
force on the cantilever is
F(vectorr) = ??U(x,y,z)?z . (6.10)
Replacing the expression for U from equation (6.9) into equation (6.10) gives
F(vectorr) =
integraldisplay
vectorM(vectorr?)? ?vectorB?M(vectorr+vectorr?)
?z d
3r?, (6.11)
which is the convolution of the magnetization of the cantilever tip and the z deriva-
tive of the field produced by the excess magnetization. Therefore the magnetization
of the cantilever tip can be regarded as a convolution factor that can be either cal-
culated numerically or found directly by measuring a device with a known magnetic
field profile. The field produced by the excess magnetization in the spin injection
device can be calculated using Green?s functions as
vectorB?M(vectorr) = ?0
4????
integraldisplay ? vectorM(vectorr?)
|vectorr?vectorr?|d
3r?, (6.12)
for any function ? vectorM, not necessarily continuous. If this expression is replaced into
equation (6.11), and the ? operators are applied, the result is
126
F(vectorr) = ??04? ??z
integraldisplay integraldisplay
d3r?d3r??
?
??3? vectorM(vectorr
?)?(vectorr+vectorr? ? vectorr?) vectorM(vectorr?)?(vectorr+vectorr? ? vectorr?)
|vectorr+vectorr? ? vectorr?|5
+?
vectorM(vectorr?)? vectorM(vectorr?)
|vectorr+vectorr? ? vectorr?|3
?
?, (6.13)
which has the same structure as the force between two dipoles. This last equation
is specially useful for estimating the magnitude of the force to be measured, while
equation (6.11) is useful for finding the convolution factor for a particular cantilever.
The magnetic field produced by the excess magnetization shown in Fig. 6.3 for a
wire of specified dimensions can be calculated directly by using equation (6.12) if
the magnetization is assumed to be uniform along the cross section of the wire. For
a 1.2 ?m wide, 51 nm thick wire, a separation of 400 nm between the upper and
lower Cu sections, and a current of 9.1 mA (see Fig. 6.3) the calculated field in the
z direction at z=200 nm as a function of x and y is shown in Fig. 6.4(a). Only
the z component of the field is relevant since the magnetization of the cantilever tip
is constrained to be in the vertical direction. Two diagonally opposed peaks and
holes are visible from the figure showing that the field in the top Cu wire is exactly
opposite as that in the lower Cu wire since they come from accumulation of different
spins. The maximum of the plotted field is about 0.57 ?T. However, from equations
(6.2) and (6.11) it is clear that in the normal MFM operation mode the cantilever
is sensitive to ?2Bz/?z2. Furthermore, it will be shown later in this chapter that in
the resonant MFM operation mode the cantilever is sensitive to ?Bz/?z, therefore
information on these quantities is relevant. Figures 6.4 (b) and (c) show the first
127
? 1 ? 0.5
0 0.5
1x (?m)
? 1
? 0.5
0
0.5
1
y (?m)
? 20
0
20?2Bz /?z2
(?T/?m2)
? 1 ? 0.5
0
0.5
1x (?m)
? 1
? 0.5
0
0.5
1
y (?m)
? 4
? 2
0
2
4
?Bz /?z
(?T/?m)
? 1 ? 0.5
0
0.5
1x (?m)
? 1
? 0.5
0
0.5
1
y (?m)
? 0.5
? 0.25
0
0.25
0.5
Bz (?T)
(a)
(b)
(c)
Figure 6.4: (a) Magnetic field in the z direction as a function of x and y at a height
of 200 nm above a 1.2 ?m wide, 51 nm thick wire carrying a current of 9 mA, due
to the excess magnetization shown in Fig. 6.3. (b) First and (c) second derivatives
of Bz with respect to z.
128
and second z derivatives of the field in the z direction, with maximum values 4
?T/?m and 37 ?T/?m2 respectively. This information will be useful for estimating
the magnitude of the force produced by the excess magnetization on the cantilever.
6.3 Increasing the resolution of the cantilever
Commercially available MFM cantilevers have a magnetic coating on all of the tip
and cantilever. This means that slowly decaying magnetic signals affect the can-
tilever response over a large scanning area. This limits the resolution since magnetic
signals coming from very distinct parts of the sample are being averaged. In other
words the convolution factor is too broad so detecting small features is impossible.
In order to enhance the resolution of the cantilevers, a magnetic coating has been
applied to nonmagnetic cantilevers on a small region near the apex of the tip. In
this section I will describe the procedure used to fabricate this high resolution tips
and will compare them with commercially available ones.
The starting point is an etched silicon probe from Veeco (model FESP), where
the cantilever and tip have already been defined. A mask, e-beam lithography, lift-
off procedure similar to the one described in chapter 4 is used, so only the new
variations in the fabrication will be discussed in detail. The probe is mounted on
a holder so that the inner part of the tip faces up. PMMA is used to cover the
whole probe except for the cantilever and tip. Then, while heating the probe to
a temperature of 100 o C, one drop of 150k PMMA is used to cover the tip and
cantilever for 15 seconds. A thin layer of PMMA sticks to the tip and cantilever,
and the rest of the PMMA drop is spun off at 3000 rpm. The probe is then baked
129
at 170 o C for an hour and the procedure is repeated with one more layer of 150k
PMMA and then with one layer of 360k PMMA. A final bake of two hours at 170 o C
is done to completely remove any remaining PMMA solvent. A low current electron
beam rasters the bottom part of the probe to find the apex of the tip. This has to
be done in a time shorter than the exposure time of the PMMA, or else the exposed
areas will be developed. Once the apex of the tip has been aligned in the center of
the field scale, the e-beam is used to expose a small region around the apex. The
probe is then developed and a film of cobalt between 50 nm and 150 nm is thermally
evaporated, followed by a thin protective chromium coverage. After lift-off, only a
small area around the apex of the tip has Co covering. Three different tips with
varying Co area are shown in Fig. 6.5. Even though by design the tips shown in (a)
and (b) were supposed to have the same magnetic covering, there is some variation
since the e-beam dose can change and since the lift-off is difficult.
In order to see the improvement of the resolution of the cantilever, the mag-
netic field produced by a straight 4.8 ?m wide line carrying a current of 19 mA
was measured in the normal MFM (phase) mode with both a normal and a high
resolution probe at a scan height of 240 nm and scan rate of 0.1 Hz. The result is
shown in Fig. 6.6. The dashed black line shows the expected response for a point
probe3, while the solid black (gray) line shows the response for a high resolution
(normal) probe. The left (right) vertical scale gives the signal measured by the
photodetector for the high resolution (normal probe). The units of the vertical axes
3Normalized to match the features of the other two curves.
130
(a)
(b)
(c)
Figure 6.5: High resolution tips with small cobalt covering near the apex.
131
Figure 6.6: Comparison of the response of the normal MFM and high resolution
probes to the field produced by a straight current carrying line. The high resolution
(normal) probe response is shown with a solid black (gray) line, while the ideal
response to a point probe is shown with a dashed black line.
are Volts, which can be converted to real units of phase (radians) after calibration
(not necessary for this comparison). The high resolution cantilever has a response
which is more than 2.5 times narrower than the normal cantilever, and the signal
clearly goes to zero within the scanning range, as opposed to the signal from the
normal cantilever which decays very slowly and on the right hand side has only
decreased by 50% from the maximum value. However, the increase in resolution is
at the expense of decreased sensitivity, as illustrated by the difference in the scale
of the response for each probe. The decrease in sensitivity of the high resolution
132
probe can be compensated by depositing thicker films of cobalt, but this makes the
lift-off process more difficult.
6.4 Resonant MFM operation
The normalmodeof MFMoperationwas described inthe first section ofthis chapter.
It was shown that the detected signal is the phase shift of the cantilever oscillatory
motion given by equation (6.4), which is proportional to the derivative of the force.
Since the magnetic force is proportional to the derivative of the field, the cantilever
gives information on ?2Bz/?z2. A variation of MFM detection can be implemented
for systems in which the magnetic field produced by the sample at a fixed position
is time dependent. In particular, for systems that produce a magnetic field that
oscillates at the natural frequency of the cantilever ?0 in a ?resonant? way. Ex-
amples of such systems are a wire carrying an oscillating current at frequency ?0,
or a rotating magnetic moment with frequency ?0. In this case the amplitude of
oscillation gives information on the force, that is, on the first derivative of the field,
?Bz/?z. In this section I will present the model for resonant MFM operation, and
will compare measurement results using both phase and resonant methods.
Using the same model described at the beginning of this chapter which lead
to equation (6.1), that is, treating the tip as a forced, damped, harmonic oscillator,
it is possible to write the equation of motion for the tip as
1
?20
?Z + 1
Q?0
?Z +Z = F0
k exp(i?0t) +
1
k
bracketleftBigg
F|Z0 + ?F?Z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingleZ
0
(Z?Z0)
bracketrightBigg
exp(i?t), (6.14)
133
where F0 is the driving force from the Z drive piezoelectric and F = F(x,y,z) is the
external force acting on the tip of the cantilever. This force is assumed to be much
smaller than the drive so that it can be expanded in a first order Taylor series about
the equilibrium position of the tip Z0 in the absence of any force. The external
force is assumed to have an explicit harmonic time dependence4 with frequency ?,
as represented by the term exp(i?t) in equation (6.14). The general solutions to
first order at frequencies ? and ?0 are given by
Z?(t) = F|Z0k ?(?)exp(??t) (6.15)
Z?0(t) = ?Qk exp(??0t), (6.16)
where ?(?) is the enhancement factor due to the nonlinear frequency response of
the cantilever, given by
?(?) =
bracketleftBigg
1? ?
2
?20 ??
?
Q?0
bracketrightBigg?1
. (6.17)
The cantilever response at the frequency ? is therefore proportional to the force.
For a magnetic interaction between the sample and tip, the force is proportional
to the derivative of the field, hence the amplitude of oscillation of the cantilever
depends linearly on ?BZ/?Z. If ? ? ?0, ? ? 1 so the amplitude of the cantilever
oscillations is equal to F/k. If, on the other hand ? ? ?0, ? ? ??20/?2 so the
amplitude response of the tip is strongly attenuated. However, if ? = ?0, ? = ?Q
4Which can be caused by an oscillating current or by a rotating magnetic moment, both of
which produce an explicitly time dependent magnetic field.
134
so the amplitude response is enhanced by a factor of Q ? 1. This means that by
oscillating the external force at the resonant frequency, the amplitude of oscillation
of the cantilever is a factor of Q larger than the amplitude when the force is time
independent.
In order to test the resonant method, the signal from a straight 4.8 ?m wide
line carrying a current of 106 ?A was measured at a scan height of 400 nm and with
a scan rate of 0.1 Hz, using both the normal (phase) and resonant methods. The
result is shown in Fig. 6.7, with a different scale for each of the traces. Since the
resonant method measures the change in amplitude, proportional to ?Bz/?z, while
the normal method measures the change in phase, proportional to ?2Bz/?z2, the
trace obtained using the resonant method has to be transformed to give directly the
second derivative of the field with respect to z. This can be done by using Fourier
transforms as shown in the Appendix [see equation (A.37)]. The scale on each axis
has been set in such a way that the maximum and minimum of each curve overlap.
The ratio between the size of the peaks and the average fluctuations (noise) is more
than two times larger for the resonant method than for the phase method. This
additional sensitivity of the resonant method is not, however, a general property,
but depends on the relative magnitude of the first and second derivatives of the field,
and on the relative size of the thermal fluctuations5 and the vibration amplitude of
the cantilever. The latter determines the effect of the thermal noise in the phase,
so for larger cantilever oscillation amplitudes, the displacements caused by thermal
5Which are the main source of noise. [70]
135
Figure 6.7: Comparison of the response of the normal MFM phase method and
the resonant method to the field produced by a straight current carrying line. The
resonant method (phase method) response is shown with a solid (dashed) line.
vibrations are relatively less important, and the phase noise is reduced. The gray
trace from Fig. 6.6 is more than two orders of magnitude larger than the dashed
trace from Fig. 6.7 since the applied current in the former measurement was about
180 times smaller than that applied in the latter one. That is why the signal to
noise ratio is much smaller in Fig. 6.7.
6.5 Force sensitivity: thermal noise
The limit in the force sensitivity of a scanning probe microscope is set by how
the thermal noise affects the zero point motion of the cantilever. Other sources
136
of noise like laser fluctuations and vibrations have been shown to be much smaller
than the thermal noise, so they can be disregarded [70]. A typical cantilever has
effectively one degree of freedom, namely, flexion in the vertical direction. All the
other flexural, torsional, and compressional degrees of freedom are effectively frozen
since their stiffness is much larger. From the energy equipartition theorem, it is clear
that at temperature T random thermal fluctuations in the oscillation amplitude of
the cantilever have an average energy k?Z2/2 = kBT/2, where kB is Boltzmann?s
constant. This means that the rms amplitude of the fluctuations is ?Z =
radicalBig
kBT/k.
This vibrational noise is the result of a noise force that excites the end of the
cantilever. The force noise is characterized by a flat power spectrum SN [29]6, which
means that the cantilever is driven with the same force at every frequency. Relating
the force noise power to the rms displacement of the cantilever requires solving
equation (6.14) replacing the external force by a driving term which is constant at
all frequencies. This can be easily done in Fourier space, where the equation can be
solved directly as
?Z(?) = ?(?)
?S
N
k , (6.18)
with ?Z(?) the Fourier transform of Z(t) and ?(?) the enhancement factor defined
in equation (6.17). The mean square amplitude of the fluctuations is given by
?Z2 = integraltext???|Z(t)|2dt. By using Parseval?s theorem
6SN, the force noise power, has units of N2/Hz, since it is the time integral of the square of
the force.
137
integraldisplay ?
??
|Z(t)|2dt=
integraldisplay ?
??
|?Z(?)|2d?, (6.19)
together with equation (6.18) and replacing ?Z2 by kBT/k , it can be shown that
the force noise in a bandwidth B, or equivalently the minimum detectable force, is
given by
Fmin =
radicalBig
SNB =
radicalBigg2kk
BT
?Qf0 B, (6.20)
wheref0 = ?0/2?. Onthe other hand, the spring constant ofa rectangular cantilever
can be approximately calculated from its length l, thickness t, and width w using
k = Ewt3/(4l3) [71], where E is the Young modulus. Therefore in terms of the
dimensions of a cantilever with a uniform cross section7, the minimum detectable
force is given by
Fmin = (?ME)1/4
parenleftBiggwt2
lQ
parenrightBigg1/2
(kBTB)1/2, (6.21)
where ?M is the density of the cantilever. Therefore the sensitivity of the cantilever
can be increased by decreasing its width and thickness or by increasing its length
and quality factor. The etched silicon cantilevers used in the experiment had ap-
proximate dimensions l = 225?m, w = 28?m, and t ? 3?m, which are typical for
commercially available cantilevers. However, the thickness of cantilevers is hard to
control during processing so the values given by the manufacturer and the actual
7For such a cantilever the resonant frequency is given by ?0 =radicalbigk/m? where m? is the effective
mass at the tip, equal to 1/4 of the mass of the cantilever.
138
measured thickness can be different. In order to solve this difficulty, the thickness
of the cantilever is taken to be unknown, so it is found in terms of the resonant
frequency and the spring constant. Therefore, if the Young?s modulus of the can-
tilever is known, the spring constant can be calculated to a good approximation by
measuring the resonant frequency of the cantilever and using the relation [71]
k = 2?3l3w
radicalBig
?3/E(f30). (6.22)
For a resonance frequency of 56.8 kHz the calculated spring constant is 1.2 N/m.
Typical values of the quality factor at atmospheric pressure are in the 140-150 range.
According to equation (6.20), at room temperature the force resolution per square
root bandwidth for such a cantilever is 19?10?15N/?Hz. This value is typical for
commercially available cantilevers at room temperature and atmospheric pressure.
Reducing the temperature to 4.2oK reduces the noise by a factor of 8, and operating
under vacuum can increase the quality factor 50 to 500 times [72], reducing the noise
by a factor between 7 and 20. Reducing the width and thickness of the cantilever
by factors of 4 and 10 respectively decreases the noise by a factor of 20. If all this
improvements are made the noise can be reduced by a factor of about 3200, giving
a force resolution of 6 attoNewtons per square root bandwidth [29].
6.6 Cantilever Calibration
The deflection of the cantilever is measured by the photodetector in units of volts.
The conversion factor ? to real units of displacement, like nanometers, depends on
139
details of the cantilever placement with respect to the laser, which can change for
every measurement. The conversion factor can be defined as ? = Z/V, where Z is
the real displacement and V is the photodetector signal. Additionally, if the signal
is to be converted to an equivalent force, it is necessary to use a model such as
that given by 6.14, which predicts that the amplitude response of the cantilever at a
frequency ? is given byZ? = ?(?)F/k. Therefore the measured force at a frequency
? is related to the photodetector voltage at that frequency by
F? = k?V?/?(?). (6.23)
This means that knowledge of the spring constant k and the conversion factor ?
are necessary to correctly scale the measured signal to units of force. In the case of
magnetic signals another calibration factor is important. In that case the magnetic
field, or the magnetic field gradient, are the quantities that need to be measured, so
a connection between them and the force felt by the cantilever has to be made. Ac-
cording to equation (6.11), this force is the convolution of the z derivative of the field
with the magnetization of the tip, which is also unknown, hence by using both (6.23)
and (6.11) it can be shown that the measured voltage at the photodetector produced
by the magnetic interaction between some field vectorB(vectorr) and the magnetization of the
tip vectorM(vectorr) is
V?(vectorr) =
integraldisplay
[ vectorM(vectorr?)?(?)/(k?)]? ?
vectorB(vectorr+vectorr?)
?z d
3r?. (6.24)
This means that the unknown conversion factor ?, spring constant, and magneti-
140
? 10 ? 5 0 5 10
x (?m)
? 2
? 1
0
1
2
?B
z/
?z
(mT
/?
m
)
? 0.6
? 0.4
? 0.2
0
0.2
0.4
0.6
F z
(mV
)
? 10 ? 5 0 5 10
x (?m)
(b)
(a)
? 10 ? 5 0 5 10
x (?m)
0
0.01
0.02
0.03
0.04
M
' 1 D
(A
m
V/
N)
Figure 6.8: (a) Position dependence of the magnetic signal measured by the can-
tilever (diamonds) at a scan height z=300 nm above the sample. Since the field is
independent of y, only the x dependence is shown. The solid line shows the convo-
lution of the expected signal [shown in (b)] with the 1D convolution factor of the
tip [shown in the inset of (a)].
141
zation of the tip can be joined as a single unknown convolution factor vectorM?(vectorr) =
vectorM(vectorr)?(?)/(k?). By measuring the cantilever response to the field produced by a
straight current carrying Cu wire of known dimensions, which can also be calculated
analytically, the convolution factor can be found by an iterative procedure known as
the ?Maximum Entropy Algorithm? [73, 74].8 First, the response of the cantilever
is measured in the x?y plane, with the tip at a given height above the sample.
This gives an image as shown by the diamonds in Fig. 6.8 (a) in units of Volts.
Only a section at constant y is shown, since the field has negligible y dependence.
The magnetic field produced by the current carrying line can be easily found using
Biot-Savart?s law. The result of the calculation of the expected signal is shown in
Fig. 6.8 (b). Experimental broadening of the features of Fig. 6.8 (b) is clear in Fig.
6.8 (a) due to the smearing effect of the magnetization of the cantilever tip, which is
not a localized dipole but is instead distributed over an area with dimensions of the
order of ?m. An ansatz for the convolution factor is given and iteratively changed
to minimize the difference between the measured signal and the convolution of the
predicted signal. A typical 1-D convolution factor is shown in the inset of Fig. 6.8
(a). To show the quality of the procedure, the calculated signal shown in 6.8 (b)
was convoluted with the 1-D convolution factor, and plotted in Fig. 6.8 (a) with a
line. The measured signal (diamonds) and the convolution of the calculated signal
with the 1-D convolution factor (line) agree very well.
8This algorithm will not be described here. One important comment, however, is that the
Maxwell equations constrain the magnetic field so that measurements of Bz in a single x?y plane
at a particular height z are enough to find Bz anywhere else in space, as long as the magnetization
and the currents remain in the x?y plane. Details can be found in the Appendix.
142
It is possible to directly obtain information about the resolution of the can-
tilever in terms of magnetic field gradient without any information about the con-
version factor ?, the spring constant, or the magnetic moment of the tip. For that
purpose it is necessary to measure the cantilever noise. This can be extracted from
the variance in the roughness of the data, which can be found by finding the dif-
ference between the real data and a smoothed version of the data. The noise as a
function of frequency for the cantilever used for the experiments is shown in Fig.
6.9. At a bandwidth of 1Hz the noise is 1?V. From Fig. 6.8 the relation between
photodetector output (mV) and field derivative (mT/?m) can be found, therefore a
noise of 1?V is equivalent to a field gradient of ? 3?T/?m. The gradient produced
by the excess magnetization for realistic values of the current, the spin relaxation
length, and the resistivity of Py and Cu, was estimated to be 4?T/?m at the begin-
ning of this chapter 9. Therefore it seems possible to detect the spin polarization of
electrons using MFM under the conditions specified before (see last footnote, and
discussion of Fig. 6.4), as long as a bandwidth equal to or smaller than 1 Hz is used.
If the cantilever noise is assumed to come completely from thermal excitations
[70], anadditionalcalibration canbemade. As shown in equation(6.20), the thermal
noise increases with the square root of the bandwidth, and for typical cantilevers
used in the experiment, it is 19 ? 10?15N/?Hz. Figure 6.9 shows the measured
noise of the cantilever in ?V, together with a linear fit with slope 0.93 ?V/?Hz.
9See Fig. 6.4(b) where the gradient produced by the excess magnetization for a 9.1 mA cur-
rent through a 1.2 ?m wide wire was calculated for the case of transparent F-N interfaces. The
calculated value of the spin current polarization in this case is approximately 5%.
143
0 1 2 3 4 5 6
BW1/2 (Hz1/2)
0
1
2
3
4
5
Noise
(?
V)
Figure 6.9: Measured cantilever noise as a function of bandwidth, with linear fit.
By comparing the expected and measured noise a scaling factor of 20 ?10?9 N/V
is found, which is valid only for signals measured at resonance.
By integrating the magnetization vectorM(vectorr), which can be found directly from the
convolution factor vectorM?(vectorr) by using values for ? (obtained from the noise calibration)
and k (measured by measuring the resonant frequency), the effective magnetic mo-
ment of the cantilever can be found to be 0.6?10?15 A m2, which is smaller than
the estimated real magnetic moment. This effective magnetic moment is not the
real magnetic moment of the tip, but the weighted moment over all possible heights,
where the weighing function is determined by how fast the magnetic field decays in
the z direction. 10 Higher order (2-D) convolution factors can be found by using not
a single current carrying line but two non-parallel lines, since in this case the field
10See equation (A.35) in the Appendix for a precise definition of the 1-D convolution factor.
144
is a function of both x and y.
145
Chapter 7
Spin injection and magnetic force microscopy detection II: Experiment
In the previous chapter I explained the basic concepts behind the idea of spin injec-
tion and MFM detection, and showed that MFM offered the resolution and sensitiv-
ity to detect the field produced by an excess magnetization created by injection of
spin polarized conduction electrons from a ferromagnetic into a nonmagnetic metal.
However, the field produced by this excess magnetization is much smaller than the
fields produced by the magnetic injector and the current, so that a new measure-
ment setup that would filter out unwanted magnetic signals had to be developed.
Additionally, a different sample fabrication procedure was used, in an effort to fab-
ricate flat Cu-Py-Cu wires with no additional topographical features at the Cu-Py
interfaces. Finally, since the cantilever is not only sensitive to magnetic fields but
also to electric fields, it was necessary to balance the potential difference between the
tip of the cantilever and the scanned region of the sample to avoid the appearance
of additional unwanted signals. I will begin this chapter by discussing the principle
of operation and the experimental setup. Then I will describe the flat sample fabri-
cation process and the balancing of the electric potential, and finally I will present
results of the measurements.
146
7.1 Principle of operation
Estimations of the size of the spin injection signal detected by the cantilever showed
that this signal is more than 200 times smaller than the signal produced by the cur-
rent circulating through the sample. The signal produced by the magnetic material
is larger but of the same order as that produced by the current, so it is neces-
sary to design a way to separate the small spin injection signal from the very large
background produced by both the current and the magnetic material. This can be
done by taking advantage of the nonlinearity of the excess magnetization in the
nonmagnetic metal, which depends on the product of the magnetization within the
magnetic material and the injected current. A magnetic injector with large aspect
ratio and small (submicron) dimensions has a very sharp hysteresis curve such that
a magnetic field applied along the easy geometrical axis will only flip the magne-
tization into either of the two easy directions. Therefore the magnetization of the
injector is expected to be in any of two possible states. If the applied magnetic
field, large enough to flip the magnetization of the injector, is reversed with a fre-
quency ?1, the magnetization of the injector will change direction with the same
frequency, vectorM = M?x(1?2?[sin(?1t)]), where ?(t) is the step function, and the easy
axis is defined to be in the x direction. If the injected current is made to oscillate
at frequency ?2 such that I(t) = Isin(?2t), the excess magnetization ?M will have
a time dependence of the form ?M ? (1?2?[sin(?1t)])sin(?2t). By expanding the
step function in a series and developing the products, it can be shown that terms of
the form cos(?1 ??2), 1/3cos(3?1 ??2), 1/5cos(5?1 ??2), ... appear. What this
147
means is that by flipping the magnetization with frequency ?1 and by injecting a
current with frequency ?2, the field from the excess magnetization can be measured
at a frequency ?1 +?2 negationslash= ?1 negationslash= ?2, so by using phase sensitive detection each of the
signals can be distinguished from each other. If the frequencies are chosen so that
?1 +?2 = ?0, the complete equation of motion of the tip can be written as
1
?20
?Z + 1
Q?0
?Z +Z = 1
k
bracketleftBigg
F0 +F?M|Z0 + ?F?M?Z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingleZ
0
(Z?Z0)
bracketrightBigg
exp(i?0t)
+ 1k
bracketleftBigg
FM|Z0 + ?FM?Z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingleZ
0
(Z?Z0)
bracketrightBigg
exp(i?1t)
+ 1k
bracketleftBigg
FI|Z0 + ?FI?Z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingleZ
0
(Z?Z0)
bracketrightBigg
exp(i?2t), (7.1)
where F?M, FM, and FI are the forces produced by the excess magnetization, the
magnetic material, and the current, expanded to first order in Z. To lowest (zeroth)
order in Z the solution to equation (7.1) at frequencies ?0, ?1, and ?2 is given by
Z0?0 = (F0 +F?M)k ?(?0) = ?Qk (F0 +F?M) (7.2)
Z0?1 = FMk ?(?1) (7.3)
Z0?2 = FIk ?(?2). (7.4)
where ?(?) is the enhancement factor defined by equation (6.17) in the previous
chapter, which is equal to ?Q at resonance. Therefore measurements at these three
different frequencies yield independent information on F?M, FM, and FI. If the
frequencies ?1 and ?2 are away from the resonant frequency ?0, the enhancement
148
factor at those frequencies will be much smaller than Q . The first order solution at
frequencies ?0 and 2?0 is given by
z1?0 = ?Qk
bracketleftBigg
z0?1?FI?z +z0?2?FM?z
bracketrightBigg
z0
(7.5)
z12?0 = z0?0 ?F?M?z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglez
0
?Q
k(2??3Q). (7.6)
Other first order solutions, which are not important, have been left out. The signif-
icance of equation (7.5) is that at frequency ?0 not only F?M is detected [equation
(7.2)] but also the derivatives of FM andFI. Therefore depending on the magnitude
of the force from the magnetic material and the current, the frequencies ?1 and ?2
should be chosen to attenuate z0?1 and z0?2 such that the largest component of z?0
comes from z0?0, that is, from F?M, instead of from z1?0.
7.2 Experimental setup
The setup that was used for the experiment is shown in Fig. 7.1. The signal from
the photo-detector, which carries the information of the cantilever deflection, is
connected to the input of three lock-in amplifiers, a Princeton Applied Research
PAR-5209 that measures the signal from FM at ?1 = 4kHz, a Stanford Research
SR-844 that measures the signal from F?M at ?0 = 56.8kHz set by the resonant
frequency of the cantilever, and a Princeton Applied Research PAR-124 that mea-
sures the signal from FI at ?2 = 60.8kHz1. The signal that drives the cantilever at
its resonant frequency (56.8 kHz in the diagram) is used to reference the Stanford
1This frequencies vary depending on the cantilever used.
149
000000000000000000000 111111111111111111111
Figure 7.1: Diagram of the experimental setup. Three lock-in amplifiers are used to
detect the cantilever response at different frequencies, to independently measure the
fields produced by the magnetic material, the current, and the excess magnetization
from spin injection.
150
SR844 lock-in and is also connected to the R input of a Mini-Circuits ZAD-8 fre-
quency mixer. The internal oscillator of the PAR-124 lock-in (set to 60.8 kHz in the
diagram) is connected to the L input of the mixer, and to a switch triggered by the
computer. The output of the switch, which is equal to the input oscillatory signal
when the switch is on and zero when the switch is off, is connected to a 200 ? resis-
tor, at which the voltage is measured, and then to one of the terminals of the sample
(the other terminal of the sample is grounded). The switch is used to acquire con-
secutive points with and without current through the sample in order to distinguish
the real effect from artifacts such as laser interference. The output of the mixer is
the superposition of two signals with frequencies 60.8kHz?56.8kHz = 4kHz and
60.8kHz+56.8kHz = 117.6kHz. After an 80kHz RC low-pass filter the higher fre-
quency component is attenuated enough so that the signal only triggers the Hewlett
Packard HP-3314A function generator at 4kHz. The TTL output of the function
generator is used to reference the PAR-5209, while its main signal output is used
to create an adjustable amplitude square wave that is amplified using an HP-6826A
to a level high enough to drive a current through the magnet. However, due to
the magnet?s finite inductance, the current is not a square wave anymore and its
magnitude is reduced. This is not crucial for the experiment if the magnetic in-
jector has a very sharp transition in the hysteresis curve, as it only introduces a
phase lag in the applied field, and if the signal can be amplified enough so that the
inductive damping is not important. Finite amplification of the electronics used in
the experiment limited the maximum frequency of the field to about 4kHz. The
151
magnet was fabricated by winding a copper wire coil on a thin nonmagnetic stain-
less steel cylindrical shell about 8mm long with a 3mm inner diameter filled with a
soft magnetic core. The coil and core were fixed to the steel shell with GE varnish.
The magnet was mounted on an aluminum support structure on the sample holder,
less than 2mm away from the magnetic injector, and held in place with Apiezon H
grease which also provided good thermal contact to the holder2. The adjustment
of the magnet position and orientation were made by eye, so even though the main
component of the field was parallel to the easy axis of the magnetic injector, small
transverse components, both in and out of plane, remain. However, if small enough,
this components do not directly affect the measurement. Calibrations of the magnet
were not very useful since the relative position between the injector and the mag-
net changed for every measured sample. However, it was experimentally seen that
the magnet was close enough to switch the magnetization of the injector in both
directions, which is the only relevant fact for the experiment.
7.3 Flat sample fabrication
In order to avoid crashing the sample and the tip, the topography of one line of
the sample is acquired, after which the tip is raised to a specified position above
the average baseline measured in the topography scan, and the magnetic response
is measured. Therefore any roughness in the topography affects the actual height
at which the magnetic scanning is being done. This means that variations in the
magnetic signal from one line to the next can appear simply because of variations
2Good thermal contact was important since the magnet heated up when driven at high currents.
152
in the topography. To avoid this problem it is desirable to fabricate completely flat
samples, or at least samples in which the topography of each line is the same (so
that all the magnetic field line scans are done in the same conditions). In the case
of a wire made of a single material this can be achieved by good lithography and
clean lift-off. However, since the samples must have interfaces between magnetic
and nonmagnetic materials, the problem gets more complicated. In principle a Cu-
Py-Cu wire as shown in Fig. 6.2 or Fig. 7.2 (f) will have uniform topography along
the y axis, solving the problem. This kind of structure can be fabricated as follows.
A permalloy film is thermally evaporated on a clean substrate. Then a bilayer of
PMMA is used to coat the wafer, and e-beam lithography followed by developing,
as explained in chapter 4, are used to define a mask as shown in Fig. 7.2(a). The
sample is placed in a vacuum chamber and ion milling is done as explained in chapter
4 to remove all the permalloy from the unprotected region [Fig. 7.2(b)]. The four
terminal resistance of a co-evaporated wafer is used as an in-situ detector to evaluate
when all of the permalloy film has been removed. Then an equivalent amount of Cu
is thermally evaporated to fill up the hole left by the milling process [Fig. 7.2(c)].
A second step of PMMA covering, e-beam lithography, and developing is done to
define the mask shown in Fig. 7.2(d). The sample is ion milled once more to trim
the Cu-Py-Cu wire to the proper width by completely removing any metal outside
the masked area [Fig. 7.2(e)]. Once again the four terminal resistance of a co-
evaporated wafer is used to decide when all the metal has been removed. The result
is a wire with a uniform cross section in y as shown in Fig. 7.2(f). However, due to
153
(d)
(e)
(f)
PMMA Mask
Copper
Permalloy
Substrate
Substrate
Substrate
PMMA Mask
Substrate
PMMA Mask
Permalloy
PMMA Mask
Permalloy
Copper
Permalloy
Substrate
Substrate
(a)
(b)
(c)
Figure 7.2: Flat sample fabrication steps. (a) A PMMA mask is defined on top of
a uniform Py film. (b) Ion milling is used to remove unprotected Py. (c) The hole
left by ion milling is filled up with Cu. (d) Second PMMA mask is defined. (e) Ion
milling is used to trim the Cu-Py-Cu line to the desired width. (f) Final result.
154
slightly different milling rates for Py and Cu, or because the evaporated Cu did not
completely fill up the hole left by the Py milling, small variations in topography are
always present, but they are smaller than with the previous fabrication technique.
7.4 Electric potential balancing
Since the tip of the cantilever is covered by a magnetic metal, the cantilever can
experience a force due to capacitive coupling with the sample. In order to minimize
this effect it is necessary to tune the voltage in the cantilever to make it as close as
possible to the voltage of the scanned area of the sample. This can be achieved by
designing the samples with an additional Cu wire in parallel with the main Cu-Py-
Cu wire as shown in Fig. 7.3. If the voltage at the top and bottom of the sample is
given by V1 and V2, the voltage in the metal at the position of the centered dotted
line is given by V0 = (V1 + V2)/2, assuming equal contact resistances at each of
the F-N interfaces. This voltage is the same at both the left and right arms of the
sample as long as the wires have a uniform cross section, even if they do not have
exactly the same width. The tip of the cantilever usually scans a region of the right
wire within 1-2 ?m of the dashed line, so if the cantilever is electrically connected
to the side wire of the left arm, the potential difference between the tip and the
scanned region of the sample will be less than 1.4 mV for current densities as high
as 2.4?1010A/m2. According to numerical and analytical calculations [75, 76] the
force between an infinite plane and the tip is about 3 ? 10?15N for the tips used
in the experiment, and for a 1.4 mV potential difference. This force should be
reduced in the case of a wire and be much smaller than the value 19?10?15 which
155
V1 V
V
1
V 20
2
0V
Permalloy
Copper
= +2
V
I
I
Figure 7.3: Schematic of the sample layout for balancing the voltage on the can-
tilever, which is connected to the point with voltage V0 on the left wire.
is the minimum detectable force at a 1Hz bandwidth due to thermal vibrations of
the cantilever. However, unbalanced electric potentials of 4mV are large enough to
produce a force equivalent to the noise, so care must be taken to distinguish any
small residual capacitive effect from the magnetic effect that is being measured. If
necessary, symmetry properties of the electrostatic interaction can help distinguish
it from the expected signal.
7.5 Results
An AFM image of a typical Cu-Py-Cu wire is shown in Fig. 7.4. Even though the
method discussed in section 7.3 was used to fabricate the sample, some variation
in the thickness can be seen. The dip corresponds to the region where permalloy
156
remains while the higher regions are made of copper.3 The width of the wire is 1.5
?m and its thickness 57 nm, even though the figure shows a bigger thickness since
part of the substrate was milled during fabrication. The thickness variation of the
wire due to the dip is about 20 nm, which causes a change in the scanning height of
the cantilever close to 10 nm, which is small compared with the scanning height of
300 nm used for the magnetic field measurements. Furthermore, since the field of
the spin polarized electrons injected into Cu is what we want to measure, only the
3Better samples were obtained but no complete set of measurements was taken for them.
Figure 7.4: Topography of the wire near the Cu-Py-Cu interfaces.
157
signal in Cu just after the injection point is relevant, so small topography changes
in the Py region are not important.
The cantilever signal at a frequency of 60.8kHz, the frequency at which current
is being driven into the sample, is shown in Fig. 7.5. The signal has the character-
istics of the field produced by a uniform current carrying wire which were observed
in the previous chapter (Fig. 6.8). A similar procedure as that used in section 6.6
can be followed to obtain the convolution factor. The current driven through the
wire is 2 mA, which gives a current density of 2.34?1010A/m2.
0 0.50
1.0 1.5
2.0
2.5x (?m)
00.10
0.20
0.30
0.40
y (?m)
? 0.1
? 0.05
0
0.05
VI (mV)
Figure 7.5: Measured cantilever response to the magnetic field produced by the
current at a frequency of 60.8 kHz.
158
The force produced by the magnetic injector is detected by measuring the
cantilever response at 4kHz, which is the frequency at which the magnetization of
the injector is being reversed. Figure 7.6 shows the measured cantilever deflection as
a function of position for the sample shown in Fig. 7.4 before using any conversion
factor or applying any deconvolution. A dipole field structure at the place of the
magnetic injector can be seen. It is important to point out that Figs. 7.5 and
7.6 give the cantilever deflection as measured by the photodetector, and not the
force. Therefore it is not correct to directly compare them to conclude that the
Figure 7.6: Measured cantilever response to the field produced by the magnetic
injector at a frequency of 4.32kHz.
159
force is similar in both cases. A correct comparison can only be made after scaling
each of the measured amplitudes by the factor?(?) defined by equation (6.17) which
characterizes the response of the system to a force F at a frequency ?, and assuming
that the 2D convolution factors are equivalent. The values of ? are -6.84+0.34? for
the force produced by the current and 1.006+0.0005? for the force produced by the
magnetic injector, which means that the force produced by the current is about
7 times smaller than that produced by the magnetic injector, even for a current
density as high as 2.34?1010A/m2.
The response of the cantilever at its resonant frequency is shown in Figs. 7.7
and 7.8, at two perpendicular phases. The in and out of phase components are
acquired directly and then, for each of them, the signals with and without current
are subtracted4. The resulting two images at phases 0o and 90o are then combined
to generate two new images at phases 15o and 105o, which are the ones presented
here. This is done since in and out of phase are not well define due to unaccounted
phase lags, and the 0o and 90o images seem to have components of the same data.
The additional rotation angle of 15o is chosen so that the images appear to be
independent.
Some features are clearly visible in Fig. 7.7, where the signal would appear
to be only about ten times smaller than that produced by the magnetic injector.
However, after scaling it by the factor ?(?0) = ?Q where the quality factor Q is 147,
4It is worth noting that points with and without current are acquired consecutively instead of
line by line so the subtraction technique is very effective in removing additional artifacts like laser
interference.
160
Figure 7.7: Measured cantilever response at a frequency of 56.8 kHz and 15 degrees
phase.
this signal is about 1500 times smaller than that produced by the magnetic injector.
This shows how the setup allows the extraction of very small signals that are buried
in much larger unwanted backgrounds, which was impossible with the standard
magnetic force microscopy setup and phase operation. The signal to noise ratio is
observed to be about 4.5. The features that can be observed in Fig. 7.7 are first, an
increased signal (red) throughout the center of the scanning area (at x=1.5?m) with
a width of about 1.5?m (the center position and width of this feature agree with the
position and width of the wire) and a size of about 18?V. And second, two regions
161
Figure 7.8: Measured cantilever response at a frequency of 56.8 kHz and 105 degrees
phase.
of decreased signal (blue) to the sides, at the y coordinate at which the magnetic
injector is found. None of these features has the correct symmetry expected for
the field produced by a decaying excess magnetization in Cu (see Fig. 6.4), so the
expected signal must be buried within the noise, and the observed features must
have a different origin.
It was pointed out in section 7.4 that the electric sensitivity of the tip was
a concern, hence the electric potential in the tip was balanced so that it would be
162
as close as possible to the electric potential in the scanning area. Additionally, the
sample was mounted on a metallic plate which was set at the same potential as
the tip, to create a uniform background electric potential. However, the red feature
in Fig. 7.7, which extends over the width and length of the metallic line, seems
to be caused by changes in the capacitance between the tip and the metallic wire
as the tip scans over the wire, together with an x dependence of the electrostatic
potential due to the presence of the wire. For a 10 mV potential difference along
the x axis, the estimated electrostatic force (see section 7.4) is about 140?10?15N.
From Fig. 7.7 the noise can be measured to be 1.77?V at a bandwidth of 1 Hz,
and assuming that it is completely thermal in origin (as done at the end of section
6.6), the calibration factor 10.7?10?9N/V can be found5. According to this factor
the size of the red feature is 193?10?15N, which is close to the estimated value,
confirming that this feature can be due to electrostatic effects since it agrees both
qualitatively and quantitatively with what is expected (within the roughness of the
estimations of the unbalanced potentials).
The other feature observed in Fig. 7.7 (the blue dips) can be explained by
equation (7.5), which shows that at the fundamental frequency not only the effect of
the excess magnetization can be detected but also a higher order signal coming from
the product of FI, FM, and their derivatives can be measured. In order to confirm
this idea, it is necessary to rewrite equation (7.5) in terms of the measured signals.
This can be done by using the special property of the magnetic fields generated by
5This factor is only valid for scaling signals that have been measured at resonance.
163
a current and an in plane magnetic moment [see equation (A.22) in the Appendix],
?Bz(x,y,z0)
?z = F
?1
x,y
bracketleftBig
Fx,y[Bz(x,y,z0)]
radicalBig
k2x +k2y
bracketrightBig
, (7.7)
which states that the derivative with respect to z of the magnetic field in the z
direction at a height z0 can be found if the field in the z direction at the same height
Bz(x,y,z0) is known, by means of Fourier transforms between the (x,y) and (kx,ky)
spaces, Fx,y6. The final expression is given by
V1?0 = ?(?0)?
bracketleftBiggV
?1
??2
?V?2
?z +
V?2
??1
?V?1
?z
bracketrightBigg
, (7.8)
where ?V/?z is calculated according to equation (7.7) [or equivalently by using
equation (A.37) in the Appendix]. The factor ? can be calculated from the noise
data as done before in section 6.6. The result of applying equation (7.8) to the
measured data (Figs. 7.5 and 7.6) is shown in Fig. 7.9. Two dips with a size of
-34?V and -60?V can be seen in the same place as those seen in Fig. 7.7. The size
of the large dip is not very reliable since in the data processing Fourier transforms
have been used, and since the data is neither periodic nor goes to zero on some of
the boundaries, significant errors can be introduced, specially on the regions close
to the borders where the data does not vanish. The signal shown in Fig. 7.5 is
close to zero (or at least close to a constant level) at x=0, but at x=3?m it is still
close to the maximum negative value (since the field has not yet decayed to zero at
this position), so large errors close to the boundary at x=3?m are expected. On
6See the Appendix for more information.
164
Figure 7.9: V1?0 generated by equation (7.8).
the other hand the data close to the other boundary is expected to be more reliable
since the magnetic signals have already decayed. Therefore a dip magnitude of -
34?V should be much closer to the real value than -60?V. From Fig. 7.7 the size
of the dips is about -23?V. Given the size of the errors that can be introduced by
the procedure of calculating V1?0 from the measured data, the difference between
the measured and calculated values can be expected. The fact that the qualitative
behavior is completely reproduced by the calculation and that the magnitude is
165
within a factor of 1.57, gives good evidence that the measurement setup is working
correctly and that the model used to interpret the results is correct. Once again,
given the accuracy with which quantities such as ? and k are known, given the noise
in the measurements, and given the systematic errors introduced by the calculations,
the model is verified by the measurements.
However, no evidence for the detection of the excess magnetization in Cu was
found. This was due in part to the thermal noise and low quality factor which limited
the sensitivity of the MFM to 19?10?15N, but also to the fact that unbalanced
electrostatic effects and higher order signals appearing from the nonlinearity of the
measurement were larger than the expected signal. The estimated size of the spin
signal was larger than the measured noise, but interface spin-flip scattering could
have decreased its size to below the noise level. This meas that the actual current
spin polarization is smaller than the expected value of 5%. Transport measurements
with tunnel barriers showed typical current spin polarizations of the order of 5%
and as high as 17%, so it seems possible that by using resistive F-N interfaces the
magnitude of the signal can be increased to a measurable value with the current
configuration. However, the existence of resistive contacts increases the problems
associated with unbalanced electrostatic potentials. Therefore decreasing the noise
level by reducing the width and thickness of the cantilever [see equation (6.21)] is
useful for detecting the spin signal. It is also useful to increase the quality factor or
use higher frequencies in order to attenuate the higher order signals [see equation
71.5 for the small dip or 2 for the average of the two dips.
166
(7.5)], increase the injected current spin polarization by using resistive barriers (Fig.
5.7), and reduce the electrostatic force between the tip and the sample.
167
Chapter 8
Summary and Conclusions
8.1 Transport
Spin injection and detection using both local and nonlocal configurations with trans-
parent and resistive interfaces was shown for Co-Cu and Py-Cu spin valve structures.
True nonlocal measurements on transparent Py-Cu spin valves were done, as op-
posed to previous measurements by Jedema et. al. [22] where their cross geometry
measurements could include artifacts such as Hall effect and anomalous magneto
resistance (AMR). The lack of reproducibility in the quality of the interfaces was
pointed out. The large confidence intervals in the fits, and the fact that quantities
not measured in the experiment such as the spin relaxation length and the spin
resolved conductivities of Py either needed to be known for fitting the data or were
used as fitting parameters, led to the conclusion that length dependence measure-
ments of the spin relaxation length in mesoscopic wires with transparent interfaces
was not reliable.
The temperature dependence of the spin related resistance was studied for the
first time. In particular, the temperature dependence in the range from 100oK to
300oK of Py-Cu and Co-Cu transparent spin valve structures in the nonlocal config-
168
uration was measured. The temperature behavior of the antisymmetric component
of the nonlocal resistance showed a linear behavior, which confirms the fact that
the spin relaxation length is proportional to the mean free time so that at higher
temperatures, where phonons are the main source of scattering, both relaxation
times are proportional to 1/T. The existence of a previously unknown component
of the nonlocal resistance, symmetric under the rotation of the magnetization direc-
tion of injector or detector, was demonstrated. This component vanished at lower
temperatures (4.2oK-100oK) but increased nonlinearly above 100oK. It was shown
that by assuming interfacial spin-flip scattering so that spin resolved currents were
not conserved at the interfaces, the existence of this new symmetric component
could be phenomenologically understood, and information about the temperature
dependence of the spin-flip scattering could be obtained. Confirmation that this
new signal also came from spin transport effects was given by comparing the length
dependence of this signal with the length dependence of the normal spin signal. Ev-
idence to verify that the symmetric spin signal was not an artifact due to geometric
effects, heating and thermopower, or current leakage, was also given. However, ques-
tions concerning the microscopic origin of the interfacial spin-flip scattering and its
temperature dependence remain.
In terms of device applications, interfacial spin-flip scattering can significantly
reduce the size of the changes in resistance in Giant Magneto Resistive (GMR) struc-
tures and Magnetic Tunnel Junctions (MTJ). In the first case this means reduced
storage capacities in hard drives and slower response times while in the second case
169
it means higher power dissipation for readout of the MTJ state and larger RC time
constants, and therefore reduced storage capacity and lower operating frequencies
for MRAM?s made of arrays of MTJ?s.
Spin precession experiments in mesoscopic Cu wires were done for the first
time. By using resistive Co-Cu interfaces and a nonlocal geometry the degree of
injected spin current polarization was found to change over a wide range, from 0%
to 17.2%. Although this high value of the spin current polarization is slightly less
than half the value in Co, such a kind of device was not reproducible. The spin
relaxation length was measured more precisely than with the length dependence
method since a single injector and detector were used, avoiding the issue of interface
reproducibility. An average value of 601 nm with a standard deviation of 6% was
found, with the statistical deviation within the measurements error coming from
uncertainties in the length and resistivity of the Cu wire segments between injector
and detector.
The contact resistance was measured directly in most of the devices. This
allowed better characterization of each contact, and verified that the equations valid
in the limit of high contact resistance could be used.
Future work in this area includes using different kinds of resistive barriers,
and different magnetic and nonmagnetic materials. Also, studying spin valves at
lower temperatures might provide information on spin relaxation due to boundaries
and impurities. In addition, the existence of spin polarized currents in the absence
of any charge current, as in the case of the studied nonlocal geometry, opens the
170
possibility of studying fundamental issues related to pure spin transport, such as the
study of spin transport statistics by measuring the zero charge current shot noise.
8.2 MFM
Spin injection into Cu using magnetic injectors, and detection of the excess mag-
netization with MFM was proposed, analyzed, and tested. It was theoretically
shown that MFM detection was possible. However in practice, due to reduced spin
injection which buried the signal below the thermal noise level, unbalanced elec-
trostatic potentials, and higher order magnetic effects which masked the signal, it
was impossible to detect the excess magnetization. However, improvements in the
cantilever resolution were made by reducing the size of the magnetic covering to a
small micron-sized region close to the apex of the tip. Improvements in the MFM
sensitivity by means of a resonant method were made, and the results were com-
pared with those from normal (phase) MFM operation by means of Fourier analysis,
using special properties of the magnetic field produced by in-plane currents.
Additionally, a multi-frequency measurement setup that could separate the
very large signals produced by the magnetic fields of injector and current from the
small signal produced by the field of the excess magnetization was tested. Other
artifacts, such as laser interference, were removed by subtracting consecutively ac-
quired data points with and without current. The measurements verified the model
used to interpret the data, and showed that the sensitivity was high enough to detect
higher order effects coming from the injector and current fields, but the excess mag-
netization signal was still smaller than those effects. By repeating the experiment at
171
higher frequencies, unwanted higher order signals could be attenuated. Increasing
the quality factor of the cantilevers by operating in vacuum could also increase the
weight of the wanted to signal with respect to the higher order unwanted effects,
as well as reduce the effects of thermal noise. In addition, reducing the noise by
changing the cantilever geometry or operating at lower temperature would enhance
the signal to noise ratio.
Reducing the electrostatic coupling between the tip and the sample or bal-
ancing the electrostatic potential so that the electric force on the tip is constant is
essential. However this is not an easy task since decreasing the electrostatic cou-
pling usually also decreases the magnetic coupling, and improving the implemented
balancing scheme seems a difficult task. It is possible that by biasing the tip at the
sum of DC and AC voltages, the electrostatic force can be measured at a particular
frequency and after data analysis subtracted from the measured signal at resonance,
to remove any electrostatic effects that appear at that frequency.
The current setup can in principle detect spin polarizations as small as P=5%.
Spin polarizations of this order were obtained by using resistive F-N interfaces,
but the transparent interfaces used for MFM measurements had lower values of P.
Therefore using resistive junctions should give values of P which are large enough
for MFM detection. However, resistive interfaces are accompanied by additional
large electrostatic effects which make the electric balancing of the sample more
difficult. MFM detection of an excess magnetization in Cu is feasible by making
small improvements as those suggested above.
172
Appendix A
Magnetic field properties, Green?s functions, and convolutions
Since in principle MFM only gives a signal related to the magnetic field in the z
direction as a function of x and y at a constant height z0, it is useful to consider
the z coordinate separately from the other two. Additionally, special properties of
the magnetic field produced by in-plane (x-y) currents or magnetizations can give
information such as the field, or even derivatives of the field, at a different height.
In this appendix I will review some of this special properties and will show that
they are still true even for convoluted magnetic signals. From now on all of the
currents and magnetizations will be assumed to have a vanishing component in the
z direction, except of course by the magnetization of the cantilever tip which will
be assumed to be in the z direction.
A.1 Magnetic field generated by an in-plane current
Let vectorJ(vectorr) = Jx(x,y,z)?x+Jy(x,y,z)?y be the current density in some bounded region
of space ?. The magnetic field in the z direction Bz(vectorr) at position vectorr outside ? is
given by
173
Bz(vectorr) = ?04??z???
integraldisplay vectorJ(vectorr?)
|vectorr?vectorr?|d
3r?
= ?04??z?
integraldisplay
vectorJ(vectorr?)? vectorr?vectorr?
|vectorr?vectorr?|3d
3r?
= ?04?
integraldisplay J
x(vectorr?)(y?y?)?Jy(vectorr?)(x?x?)
|vectorr?vectorr?|3 d
3r?, (A.1)
where as usual integrations without limits mean integration over all space. By using
the 2D Fourier transform
f(kx,ky) = 12?
integraldisplay
F(x,y)exp[?(kxx+kyy)]dxdy, (A.2)
it is possible to rewrite equation (A.1) as
bz(kx,ky;z) = ?08?2
integraldisplay J
x(vectorr?)(y?y?)?Jy(vectorr?)(x?x?)
|vectorr?vectorr?|3 exp
?(kxx+kyy)dxdyd3r?. (A.3)
Replacing x?x? ?x and y?y? ?y equation (A.3) can be written as
bz(kx,ky;z) =
integraldisplay
dz?
bracketleftBigg
(A.4)
?0
8?2
integraldisplay
Jx(vectorr?)exp?(kxx?+kyy?)dx?dy?
integraldisplay y
[x2 +y2 + (z?z?)2]3/2 exp
?(kxx+kyy)dxdy?
?0
8?2
integraldisplay
Jy(vectorr?)exp?(kxx?+kyy?)dx?dy?
integraldisplay x
[x2 +y2 + (z?z?)2]3/2 exp
?(kxx+kyy)dxdy
bracketrightBigg
,
where the integrals over dx?dy? can be identified as the 2-D Fourier transforms of Jx
and Jy,
jx(kx,ky;z?) = 12?
integraldisplay
Jx(vectorr?)exp?(kxx?+kyy?)dx?dy?
174
jy(kx,ky;z?) = 12?
integraldisplay
Jy(vectorr?)exp?(kxx?+kyy?)dx?dy?, (A.5)
and the integrals overdxdy can be identified as the Fourier transforms of the Green?s
functions,
gy(kx,ky;z?z?) = 12?
integraldisplay y
[x2 +y2 + (z?z?)2]3/2 exp
?(kxx+kyy)dxdy
gx(kx,ky;z?z?) = 12?
integraldisplay x
[x2 +y2 + (z?z?)2]3/2 exp
?(kxx+kyy)dxdy. (A.6)
Therefore equation A.4 can be written as
bz(kx,ky;z) = ?02
integraldisplay
[jx(kx,ky;z? )gy(kx,ky;z?z?)
?jy(kx,ky;z?)gx(kx,ky;z?z?)]dz?. (A.7)
For currents without any component in the ?z direction, the continuity equation
?Jx
?x +
?Jy
?y = 0 (A.8)
implies that in 2-D Fourier space
jxkx +jyky = 0, (A.9)
so equation (A.7) can be expressed as
bz(kx,ky;z) = ??02
integraldisplay
jy(kx,ky;z?)[gy(kx,ky;z?z?)ky/kx +gx(kx,ky;z?z?)]dz?.
(A.10)
175
If it is further assumed that the current is nonzero only in a slab between z = 0 and
z = d and that inside the slab the current does not have any z dependence, then
bz(kx,ky;z) = ??02 jy(kx,ky)
integraldisplay d
0
[gy(kx,ky;z?z?)ky/kx +gx(kx,ky;z?z?)]dz?,
(A.11)
which after a change of variables can be formulated as
bz(kx,ky;z) = ??02 jy(kx,ky)[g(kx,ky;z)?g(kx,ky;z?d)], (A.12)
with the definition
g(kx,ky;z) =
integraldisplay z
[gy(kx,ky;z?)ky/kx +gx(kx,ky;z?)]dz?. (A.13)
Since the Fourier transforms applied for this derivation are only from the x-y to the
kx-ky space, equation (A.12) can be simply extended to the derivatives of the field
by directly applying to it the operator ?/?z:
b?z(kx,ky;z) = ??zbz(kx,ky;z). (A.14)
Now, by replacing xexp(?kxx) ??/?(?kx)exp(?kxx) and similarly for yexp(?kyy) in
equations (A.6), and then changing the integration from rectangular (x,y) to polar
(?,?) coordinates, gx(kx,ky;z) and gy(kx,ky;z) can be expressed as
gx(kx,ky;z) = 12?? ??k
x
g0(z)
176
gy(kx,ky;z) = 12?? ??k
y
g0(z), (A.15)
with
g0(z) =
integraldisplay
?exp(?k?cos?)(?2 +z2)3/2 d?d?, (A.16)
and k =
radicalBig
k2x +k2y. By direct integration g0(z) is found to be
g0(z) = 2?exp?k|z|/|z|, (A.17)
so replacing this result in (A.15) gives
gx(kx,ky;z) = ?kx?k exp?k|z|
gy(kx,ky;z) = ?ky?k exp?k|z|, (A.18)
and therefore, from (A.13),
g(kx,ky;z) = ? k?k
x
integraldisplay z
exp?k|z?|dz? = 1?k
x
exp?k|z|. (A.19)
Finally, for z>d, (A.12) can be expressed as
bz(kx,ky;z) = ? ?02?k
x
bracketleftBig
1?expkd
bracketrightBig
jy(kx,ky)exp?kz. (A.20)
Since the z dependence is separable from the kx,ky dependence, it is clear that if
the field is known at a height z, then it can be found at a height z? by using
177
bz(kx,ky;z?) = bz(kx,ky;z)exp?k(z??z), (A.21)
and then making a 2-D inverse Fourier transformation. Additionally, by applying
the z derivative operator as shown in (A.14), it is clear that the derivative of the
field and the field are related by
b?z(kx,ky;z) = ?kbz(kx,ky;z). (A.22)
The n?th order derivative can be easily found to be related to the field by
b(n)z (kx,ky;z) = (?k)nbz(kx,ky;z). (A.23)
A.2 Magnetic field generated by an in-plane magnetization
Let vectorM(vectorr) = Mx(x,y,z)?x+My(x,y,z)?ybe the magnetization in some bounded region
of space ?. The magnetic scalar potential ?(vectorr) at position vectorr outside ? is given by
?(vectorr) = ??04???
integraldisplay vectorM(vectorr?)
|vectorr?vectorr?|d
3r?
= ??04?
integraldisplay M
x(vectorr?)(x?x?) +My(vectorr?)(y?y?)
|vectorr?vectorr?|3 d
3r?. (A.24)
As usual, the relation between the magnetic scalar potential and the magnetic field
is given by vectorB = ??? so Bz = ???/?z. The procedure leading to equation (A.7)
in the previous section can be followed to give
178
?(kx,ky;z) = ??02
integraldisplay
[mx (kx,ky;z?)gx(kx,ky;z?z?)
+my(kx,ky;z?)gy(kx,ky;z?z?)]dz?, (A.25)
where gx and gy are given by equation (A.6) and mx and my are the 2-D Fourier
transforms of the magnetization1 given by
mx(kx,ky;z?) = 12?
integraldisplay
Mx(vectorr?)exp?(kxx?+kyy?)dx?dy?
my(kx,ky;z?) = 12?
integraldisplay
My(vectorr?)exp?(kxx?+kyy?)dx?dy?. (A.26)
Since there is no continuity equation for the magnetization, it is necessary to keep
both independent functions mx and my in the procedure. If the magnetization is
further assumed to be nonzero only in a slab between z = 0 and z = d, and have
no z dependence in the slab, the procedure which lead to equation (A.20) in the
previous section can be followed identically to give
bz(kx,ky;z) = ??02?k
bracketleftBig
1?expkd
bracketrightBig
[kxmx(kx,ky) +kymy(kx,ky)]exp?kz, (A.27)
after using bz = ???/?z. Since once more thez dependence is completely separable
and has the same form as in (A.20), equations (A.21)-(A.23) also apply to the case
of a slab with in plane magnetization.
1Not to be confused with the magnetic moment, which is usually also denoted by m.
179
A.3 2-D Convolution
The signal detected by the MFM is not the derivative of the field in the z direction
but instead its convolution with the magnetization of the tip. Therefore it is not
obvious that the relations derived above, which apply to the field Bz, should also
hold for the measured signal. In this section it will be shown that equations (A.21)-
(A.23) are also valid for the measured convoluted signal.
Let U be the negative of the convolution of the magnetic field with the mag-
netization vectorM = M?z of the tip, assumed to be in the z direction,
U(vectorr) = ?
integraldisplay
Bz(vectorr+vectorr?)M(vectorr)d3r?. (A.28)
U is therefore the magnetic energy stored in the system. The measured signal, up to
calibration constants, is given by V = ??U/?z. Following the procedure described
at the beginning of section A, it is possible to write the 2-D fourier transform of U
as
u(kx,ky;z) = ?2?
integraldisplay
bz(kx,ky;z+z?)m(?kx,?ky,z?)dz?. (A.29)
Using the property (A.21) of the magnetic field, u can be written as
u(kx,ky;z) = ?bz(kx,ky;z)m2D(?kx,?ky), (A.30)
where the 2-D convolution factor in Fourier space m2D(kx,ky) is given by
180
m2D(kx,ky) = 2?
integraldisplay
exp?kz? m(kx,ky,z?)dz?. (A.31)
Therefore the measured signal v is given by
v(kx,ky;z) = b?z(kx,ky;z)m2D(?kx,?ky), (A.32)
Since the z dependence of v(kx,ky;z) is given exactly by that of b?z(kx,ky;z), it is
clear that equations (A.21)-(A.23) also apply to v(kx,ky;z), hence
v(kx,ky;z?) = v(kx,ky;z)exp?k(z??z) (A.33)
v(n)(kx,ky;z?) = (?k)nv(kx,ky;z). (A.34)
A.4 1-D Convolution
If the measured field is constant in the y direction, then a 1-D Fourier transform can
be used instead, and the procedure of the last section can be repeated. In this case,
however, the measured signal is v(kx;z) and the 1-D convolution factor is given by
m1D(kx) = ?2?
integraldisplay
exp?kz? m(kx,y?,z?)dy?dz?. (A.35)
Equations (A.33) and (A.33) are replaced by
v(kx;z?) = v(kx;z)exp?k(z??z) (A.36)
v(n)(kx;z?) = (?k)nv(kx;z). (A.37)
181
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